Results of the Numerical Solution of the Laplace Equation for Solids of Revolution with Shapes from - 0.0025 to - 0.2500

V. I. MELIK-GAIKAZYAN

Results of the Numerical Solution of the Laplace Equation for Solids of Revolution with Shapes from - 0.0025 to - 0.2500

V. I. MELIK-GAIKAZYAN

Submitted 1965 | SovietRxiv: ru-196501.14490 | Translated from Russian

Abstract Generated abstract

This paper presents numerical solutions of the Laplace capillarity equation for axisymmetric interfaces with negative shape parameters, addressing a gap in earlier Bashforth, Adams, and Fordham tables for small pendant drops and sessile bubbles. Using arc length as the independent variable for negative values of the dimensionless parameter beta, calculations were performed on a Minsk-1 digital computer for shapes ranging approximately from beta = -0.0025 to beta = -0.2500. The resulting tables give dimensionless curvature, angle, coordinates, volume, and surface area, enabling calculation of surface tension and geometrical parameters for small bubbles and drops relevant to flotation, electrochemistry, adsorption, spreading, and contact-angle studies.

Full Text

RESULTS OF THE NUMERICAL SOLUTION OF THE LAPLACE EQUATION FOR BODIES OF REVOLUTION WITH SHAPES FROM −0.0025 TO −0.2500

V. I. MELIK-GAIKAZYAN

(Presented by the Scientific-Methodological Seminar of the KhTF)

The Laplace equation (1806) describes the interface between two liquids, or between a liquid and a gas, acted upon only by capillary and gravitational forces. It makes it possible to express the surface tension in terms of the densities of the bounding phases and the magnitudes of the principal radii of curvature at two points of the surface located at different levels. In the special case of bodies of revolution with a vertical axis of symmetry, one can confine oneself to one radius of curvature at the point of the surface whose normal coincides with the axis of revolution, and to the coordinates of any point of the surface located at another level. In this case, which is considerably more convenient for measurements, the equation can be solved only approximately. The most accurate numerical solution of the Laplace equation for this case was given in 1883 by Bashforth and Adams [1], by the method of the latter [2]. It was carried out in order to verify the mathematical theory of capillarity by comparing the computed and actual forms of meridional sections of mercury drops of various sizes lying on a smooth horizontal surface. The agreement proved to be complete, and the computed, very accurate tables formed the basis of many absolute methods for determining surface tension under static conditions from the shapes of sessile or pendant drops and bubbles—for example, of viscous oils and resins, molten metals, alloys and silicates, solutions, and other interfaces. In addition, the tables of Bashforth and Adams were used to compile correction tables and relations in order to take into account the influence of the weight of the liquid in the meniscus on the change of its curvature with height, which is necessary in the precise determination of surface tension [3], for example, by the method of capillary rise of a liquid or by the method of the maximum pressure of a gas (liquid) in a bubble (drop).

The use of these tables for solving one flotation problem [4] proved very useful; however, it showed that, for the case of sitting bubbles or pendant drops about one millimeter in size, which are of interest for froth flotation and electrochemistry on drop electrodes, the tables of Bashforth and Adams [1] and the supplementary tables of Fordham [5] do not contain the initial data needed for the calculations. To fill this gap, about one hundred numerical solutions of the Laplace equation were carried out on an electronic digital ...

on the TPI “Minsk-1” machine*). Some of the results obtained are presented in the form of tables in the present article. In the tables the generally accepted notation is used; it follows from the Laplace equations, written by Adams [1, 3, 5] in the following form, transformed for solution:

\[ \frac{1}{\rho/b}+\frac{\sin\varphi}{x/b}=2+\beta z/b, \tag{1} \]

where

\[ \beta=\frac{g(D_1-D_2)b^2}{\sigma}. \tag{2} \]

In equations (1) and (2) and in Fig. 1 the following notation is adopted: $x$ and $z$ are the coordinates of an arbitrary point $A$ on the surface of the body of revolution with axis $OZ$ and origin at point $O$; $\varphi$ is the angle between the normal at point $A$ and the axis of revolution; $\rho$ is the principal radius of curvature of the surface at point $A$, lying in the plane of the meridional section of the body; $S$ is the length of the arc from the origin of coordinates to point $A$; $D_1$ and $D_2$ are the densities of the bordering phases; $g$ is the acceleration due to gravity; $b$ is the radius of curvature of the surface at point $O$, which is chosen as the unit of scale; $\beta$ is a dimensionless quantity whose value and sign uniquely characterize the shape of the surface.

$Fig. 1. Shape of the surface with $\beta=-0.2000$, with explanations of the notation used in the Laplace equation and in the tables.$

Fig. 1. Shape of the surface with $\beta=-0.2000$, with explanations of the notation used in the Laplace equation and in the tables.

The tables of Bashforth and Adams are compiled from the dimensionless values of the abscissae $x/b$, coordinates $z/b$, and volumes $v/b^3$ for prescribed $\beta$ and $\varphi$ or $S$, with the values of $\beta$ positive for lying drops, when $D_1>D_2$, and negative for pendant drops and sessile bubbles (Fig. 1),

* The programming of the operations for the numerical solution of the Laplace equation and the correction of the results obtained according to the procedure compiled for this purpose [1] was carried out by N. F. Bezmaternykh.

when $D_1 < D_2$ and are equal to zero for a sphere. The dimensionless nature of the numbers in the tables makes them very convenient and applicable to interfaces of practically any dimensions, surface tensions, and densities of the bordering phases.

As the independent variable in solving the Laplace equation, in the case of positive $\beta$ the quantity $\varphi$ is chosen, and in the case of negative $\beta$, because of the presence of inflection points on the surface, the arc length $S$ [1] is chosen, also expressed in the form of the dimensionless quantity $s/b$.

The tables contained in this article also have columns with the values $b/\rho$ and $P/b^2$, which make it possible, respectively, to calculate readily, for the forms considered, the radii of curvature of the meridional section at any point of the surface and the areas of the curvilinear surface enclosed between point $O$ (Fig. 1) and a horizontal plane drawn at a specified level $z/b$.

Figure 2a shows the shapes of air bubbles in pure water, calculated from the tables of Bashforth and Adams and the tables of Fordham, and Fig. 2b shows some of the newly calculated shapes.

From a comparison of the curves in Figs. 2a and 2b it is clear that the previously published tables [1, 5] contained a wholly insufficient amount of data for calculating the surface tensions, volumes, areas of curvilinear surface, and other parameters of small pendant drops and sessile bubbles, which are of interest when considering adsorption and spreading phenomena of various surface-active substances on them, and also made it impossible to compare the calculated and actual contact angles at the base of the surfaces considered.

$Fig. 2. Some forms of gas bubbles in pure water at 25°C, calculated by the numerical solution of the Laplace equation: a) by Bashforth and Adams (1883) and Fordham (1948), and b) newly calculated values, the initial values—$\beta$, not marked in the figure, being respectively equal to $-0.0025$, $-0.0050$, $-0.0100$, $-0.0200$, $-0.0300$, and $-0.0400$.$

Fig. 2. Some shapes of gas bubbles in pure water at $25^\circ\mathrm{C}$, calculated by numerical solution of the Laplace equation: a) by Bashforth and Adams (1883) and Fordham (1948), and b) newly calculated values, with the initial values—$\beta$, not marked in the figure, respectively equal to $-0.0025$, $-0.0050$, $-0.0100$, $-0.0200$, $-0.0300$, and $-0.0400$.

and sessile bubbles, which are of interest when considering adsorption and spreading phenomena of various surface-active substances on them, and also precluded comparison of the calculated and actual contact angles at the base of the surfaces under consideration.

REFERENCES

F. Bashforth and J. C. Adams, “An attempt to test the theories of capillary action by comparing the theoretical and measured forms of drops of fluids,” Cambridge, 1883.
A. N. Krylov. Lectures on Approximate Calculations, Moscow, 1950.
N. K. Adam. The Physics and Chemistry of Surfaces, Moscow, 1947.
V. I. Melik-Gaikazyan and A. A. Baichenko. Investigation of the mechanism of strengthening the contact between a bubble and a coal particle by an apolar reagent, Doklady Akademii Nauk SSSR, 136, 1403, 1961.
S. Fordham, On the calculation of surface tensions from measurements of pendant drops, Proceedings of the Royal Society, A, 194, No. 1036, p. 1, 1948.

TABLE I

\[ \beta = -0,0025 \]

$\dfrac{S}{b}$	$\dfrac{b}{\rho}$	$\varphi$, radians	$\varphi$, deg. min. sec.	$\dfrac{x}{b}$	$\dfrac{z}{b}$	$\dfrac{V}{b^3}$	$\dfrac{\Pi}{b^2}$
0,1	0,999991	0,099999	5 43 46	0,099833	0,004996	0,0001	0,0313
0,2	0,999963	0,199997	11 27 32	0,198669	0,019933	0,0011	0,1252
0,3	0,999916	0,299991	17 11 17	0,295521	0,044663	0,0060	0,2805
0,4	0,999851	0,399980	22 55 17	0,389419	0,078937	0,0186	0,4960
0,5	0,999769	0,499961	28 38 44	0,479427	0,122413	0,0447	0,7691
0,6	0,999670	0,599933	34 22 25	0,564648	0,174655	0,0890	1,0974
0,7	0,999552	0,699894	40 06 35	0,644228	0,235142	0,1594	1,4775
0,8	0,999422	0,799843	45 49 39	0,717376	0,303267	0,2574	1,9056
0,9	0,999275	0,899778	51 33 12	0,783361	0,378352	0,3900	2,3775
1,0	0,999112	0,999697	57 16 42	0,841526	0,459644	0,5592	2,8884
1,1	0,998933	1,099600	63 00 08	0,891293	0,546333	0,7652	3,4332
1,2	0,998744	1,199484	68 43 31	0,932166	0,637553	1,0028	4,0065
1,3	0,998537	1,299348	74 26 49	0,963741	0,732393	1,2740	4,6026
1,4	0,998324	1,399191	80 10 38	0,985703	0,829909	1,5609	5,2155
1,5	0,998092	1,499012	85 53 13	0,997838	0,929128	1,8686	5,8392
1,6	0,997844	1,598809	91 36 18	1,000025	1,029062	2,1849	6,4674
1,7	0,997580	1,698580	97 19 17	0,992246	1,128718	2,4963	7,0938
1,8	0,997295	1,798324	103 02 10	0,974581	1,227103	2,7947	7,7122
1,9	0,996983	1,898038	108 44 58	0,947208	1,323240	3,0737	8,3164
2,0	0,996634	1,997719	114 27 39	0,910402	1,416126	3,3256	8,9005
2,1	0,996230	2,097363	120 10 12	0,864532	1,504988	3,5476	9,4585
2,2	0,995750	2,196963	125 52 36	0,810056	1,588798	3,7331	9,9850
2,3	0,995151	2,296509	131 34 48	0,747518	1,666778	3,8829	10,4748
2,4	0,994364	2,395987	137 16 47	0,677542	1,738158	3,9957	10,9228
2,5	0,993251	2,495371	142 58 27	0,600826	1,802240	4,0788	11,3248
2,6	0,991549	2,594618	148 39 38	0,518134	1,858399	4,1343	11,6766
2,7	0,988656	2,693642	154 20 35	0,430289	1,906097	4,1693	11,9748
2,8	0,982999	2,792260	159 59 49	0,338166	1,944893	4,1872	12,2164
2,9	0,969185	2,889985	165 35 21	0,242681	1,974468	4,1958	12,3990
3,0	0,917819	2,984992	171 01 38	0,144785	1,994589	4,1980	12,5208
3,07	0,703400	3,044109	174 25 52	0,075349	2,003477	4,1962	12,5692

TABLE I

\[ \beta = -0{,}0050 \]

$S/b$	$b/\rho$	$\varphi$, radians	$\varphi$, deg. min. sec.	$x/b$	$z/b$	$V/b^3$	$\Pi/b^2$
0,1	0,999981	0,099999	5 43 46	0,099833	0,004996	0,0001	0,0313
0,2	0,999925	0,199995	11 27 31	0,198669	0,019933	0,0011	0,1252
0,3	0,999832	0,299983	17 11 15	0,295521	0,044662	0,0060	0,2806
0,4	0,999702	0,399960	22 54 57	0,389419	0,078934	0,0188	0,4959
0,5	0,999539	0,499922	28 38 36	0,479430	0,122408	0,0447	0,7691
0,6	0,999339	0,599867	34 22 11	0,564652	0,174646	0,0898	1,0974
0,7	0,999107	0,699789	40 05 41	0,644238	0,235126	0,1593	1,4775
0,8	0,998842	0,799687	45 49 07	0,717394	0,303242	0,2585	1,9056
0,9	0,998547	0,899556	51 32 26	0,783393	0,378314	0,3917	2,3775
1,0	0,998221	0,999395	57 15 40	0,841580	0,459592	0,5607	2,8884
1,1	0,997866	1,099199	62 58 46	0,891377	0,546263	0,7652	3,4333
1,2	0,997486	1,198967	68 41 44	0,932292	0,637464	1,0047	4,0067
1,3	0,997078	1,298696	74 24 35	0,963922	0,732286	1,2731	4,6029
1,4	0,996644	1,398382	80 07 16	0,985956	0,829786	1,5650	5,2159
1,5	0,996182	1,498023	85 49 49	0,998179	0,928995	1,8723	5,8398
1,6	0,995690	1,597617	91 32 12	1,000475	1,028927	2,1856	6,4682
1,7	0,995162	1,697160	97 14 24	0,992826	1,128592	2,4976	7,0949
1,8	0,994592	1,796648	102 56 25	0,975313	1,227005	2,7973	7,7137
1,9	0,993968	1,896077	108 38 13	0,948114	1,323192	3,0773	8,3185
2,0	0,993269	1,995439	114 19 48	0,911505	1,416206	3,3308	8,9032
2,1	0,992467	2,094727	120 01 08	0,865853	1,505131	3,5519	9,4620
2,2	0,991509	2,193928	125 42 10	0,811614	1,589095	3,7387	9,9894
2,3	0,990321	2,293022	131 22 49	0,749331	1,667277	3,8883	10,4802
2,4	0,988755	2,391979	137 03 01	0,679622	1,738920	4,0036	10,9295
2,5	0,986551	2,490752	142 42 34	0,603183	1,803331	4,0874	11,3328
2,6	0,983189	2,589252	148 21 11	0,520771	1,859901	4,1442	11,6862
2,7	0,977509	2,687313	153 58 18	0,433204	1,908111	4,1788	11,9861
2,8	0,966463	2,784581	159 32 41	0,341352	1,947548	4,1977	12,2297
2,9	0,939799	2,880115	165 01 06	0,246125	1,977949	4,2052	12,4143
3,0	0,843059	2,970469	170 11 43	0,148472	1,999329	4,2088	12,5384
3,08	0,297495	3,024046	173 15 54	0,069277	2,010578	4,2064	12,5932

TABLE I

\[ \beta=-0{,}0025 \]

$\dfrac{S}{b}$	$\dfrac{b}{\rho}$	$\varphi$, radians	$\varphi$, deg. min. sec.	$\dfrac{x}{b}$	$\dfrac{z}{b}$	$\dfrac{V}{b^3}$	$\dfrac{\Pi}{b^2}$
0,1	0,999972	0,099999	5 43 46	0,099833	0,004996	0,0001	0,0313
0,2	0,999888	0,199992	11 27 31	0,198669	0,019933	0,0011	0,1252
0,3	0,999749	0,299975	17 11 14	0,295521	0,044661	0,0061	0,2806
0,4	0,999555	0,399940	22 54 53	0,389421	0,078933	0,0189	0,4960
0,5	0,999309	0,499884	28 38 28	0,479432	0,122404	0,0447	0,7691
0,6	0,999008	0,599800	34 21 57	0,564657	0,174637	0,0899	1,0974
0,7	0,998661	0,699684	40 05 20	0,644248	0,235110	0,1593	1,4775
0,8	0,998263	0,799530	45 48 34	0,717413	0,303217	0,2589	1,9056
0,9	0,997818	0,899334	51 31 40	0,783426	0,378277	0,3924	2,3775
1,0	0,997330	0,999092	57 14 37	0,841634	0,459539	0,5615	2,8885
1,1	0,996801	1,098799	62 57 23	0,891462	0,546193	0,7660	3,4334
1,2	0,996230	1,198451	68 39 58	0,932419	0,637375	1,0051	4,0068
1,3	0,995619	1,298044	74 22 20	0,964104	0,732179	1,2733	4,6031
1,4	0,994969	1,397573	80 04 30	0,986208	0,829663	1,5647	5,2163
1,5	0,994274	1,497036	85 46 25	0,998521	0,928861	1,8730	5,8404
1,6	0,993537	1,596427	91 28 06	1,000925	1,028791	2,1866	6,4690
1,7	0,992745	1,695741	97 09 31	0,993406	1,128466	2,4988	7,0961
1,8	0,991892	1,794974	102 50 39	0,976045	1,226906	2,7991	7,7158
1,9	0,990954	1,894116	108 31 29	0,949021	1,323142	3,0804	8,3206
2,0	0,989910	1,993161	114 11 58	0,912608	1,416234	3,3341	8,9059
2,1	0,988711	2,092093	119 52 05	0,867174	1,505270	3,5555	9,4655
2,2	0,987284	2,190896	125 31 44	0,813173	1,589388	3,7427	9,9938
2,3	0,985508	2,289539	131 10 51	0,751144	1,667773	3,8937	10,4856
2,4	0,983175	2,387979	136 49 16	0,681703	1,739675	4,0103	10,9361
2,5	0,979902	2,486143	142 26 43	0,605541	1,804415	4,0952	11,3409
2,6	0,974926	2,583903	148 02 48	0,523410	1,861394	4,1527	11,6958
2,7	0,966552	2,681017	153 36 39	0,436124	1,910111	4,1882	11,9975
2,8	0,950377	2,776962	159 06 29	0,344546	1,950183	4,2077	12,2430
2,9	0,917765	2,870385	164 27 39	0,249581	1,981400	4,2167	12,4298
3,0	0,774994	2,956401	169 23 21	0,152180	2,003910	4,2195	12,5561
3,08	0,052792	2,999660	171 52 04	0,073198	2,016594	4,2176	12,6127

TABLE I

\[ \beta = -0,0100 \]

$S/b$	$b/\rho$	$\varphi$, radians	$\varphi$, deg. min. sec.	$x/b$	$z/b$	$V/b^3$	$n/b^2$
0,1	0,999963	0,099999	5 43 46	0,099833	0,004996	0,0001	0,0313
0,2	0,999851	0,199990	11 27 30	0,198670	0,019933	0,0011	0,1252
0,3	0,999664	0,299966	17 11 12	0,295521	0,044661	0,0051	0,2806
0,4	0,999407	0,399920	22 54 49	0,389421	0,078931	0,0188	0,4960
0,5	0,999076	0,499845	28 38 20	0,479434	0,122399	0,0450	0,7691
0,6	0,998679	0,599733	34 21 43	0,564662	0,174628	0,0899	1,0974
0,7	0,998213	0,699578	40 04 58	0,644258	0,235095	0,1596	1,4775
0,8	0,997681	0,799373	45 48 02	0,717432	0,303192	0,2594	1,9057
0,9	0,997093	0,899113	51 30 55	0,783459	0,378239	0,3919	2,3776
1,0	0,996441	0,998790	57 13 35	0,841688	0,459486	0,5614	2,8885
1,1	0,995734	1,098399	62 56 01	0,891546	0,546123	0,7665	3,4335
1,2	0,994975	1,197935	68 38 11	0,932545	0,637286	1,0048	4,0070
1,3	0,994160	1,297392	74 20 06	0,964285	0,732072	1,2732	4,6034
1,4	0,993292	1,396765	80 01 43	0,986461	0,829540	1,5654	5,2167
1,5	0,992371	1,496048	85 43 02	0,998862	0,928727	1,8722	5,8410
1,6	0,991384	1,595236	91 24 01	1,001375	1,028654	2,1874	6,4699
1,7	0,990332	1,694323	97 04 39	0,993986	1,128340	2,4997	7,0972
1,8	0,989194	1,793300	102 44 54	0,976776	1,226806	2,8005	7,7169
1,9	0,987949	1,892158	108 24 45	0,949928	1,323092	3,0817	8,3226
2,0	0,986558	1,990884	114 04 09	0,913712	1,416260	3,3365	8,9086
2,1	0,984962	2,089462	119 43 02	0,868496	1,505408	3,5595	9,4689
-2,2	0,983066	2,187867	125 21 20	0,814732	1,589678	3,7475	9,9982
2,3	0,980711	2,286060	130 58 53	0,752958	1,668265	3,8996	10,4911
2,4	0,977622	2,383985	136 35 32	0,683786	1,740426	4,0174	10,9428
2,5	0,973301	2,481544	142 10 55	0,607901	1,805492	4,1029	11,3489
2,6	0,966752	2,578572	147 44 28	0,526053	1,862878	4,1613	11,7054
2,7	0,955781	2,674750	153 15 06	0,439049	1,912098	4,1977	12,0089
2,8	0,934724	2,769403	158 40 30	0,347748	1,952799	4,2179	12,2562
2,9	0,885008	2,860791	163 54 40	0,253050	1,984822	4,2272	12,4451
3,0	0,712967	2,942758	168 36 27	0,155907	2,008435	4,2303	12,5737
3,09	0,496142	2,974146	170 24 21	0,067290	2,024126	4,2276	12,6368

TABLE I

\[ \beta = -0{,}0125 \]

$S/b$	$b/p$	$\varphi$, radians	$\varphi$, deg. min. sec.	$x/b$	$z/b$	$V/b^3$	$\Pi/b^2$
0,1	0,999955	0,099998	5 43 46	0,099833	0,004996	0,0001	0,0313
0,2	0,999814	0,199987	11 27 30	0,198670	0,019933	0,0012	0,1252
0,3	0,999582	0,299958	17 11 11	0,295521	0,044660	0,0061	0,2806
0,4	0,999258	0,399900	22 54 45	0,389422	0,078929	0,0189	0,4959
0,5	0,998846	0,499806	28 38 12	0,479436	0,122395	0,0449	0,7691
0,6	0,998342	0,599666	34 21 30	0,564665	0,174620	0,0900	1,0974
0,7	0,997765	0,699473	40 04 37	0,644267	0,235079	0,1597	1,4775
0,8	0,997102	0,799216	45 47 30	0,717450	0,303167	0,2598	1,9056
0,9	0,996364	0,898890	51 30 09	0,783491	0,378202	0,3924	2,3776
1,0	0,995552	0,998487	57 12 33	0,841742	0,459434	0,5614	2,8885
1,1	0,994669	1,097998	62 54 38	0,891631	0,546053	0,7664	3,4335
1,2	0,993718	1,197418	68 36 25	0,932671	0,637198	1,0051	4,0072
1,3	0,992702	1,296740	74 17 52	0,964466	0,731965	1,2735	4,6037
1,4	0,991618	1,395956	79 58 57	0,986713	0,829417	1,5653	5,2171
1,5	0,990464	1,495060	85 39 38	0,999203	0,928593	1,8731	5,8415
1,6	0,989235	1,594046	91 19 56	1,001825	1,028512	2,1879	6,4707
1,7	0,987920	1,692904	96 59 47	0,994565	1,128212	2,5003	7,0984
1,8	0,986500	1,791626	102 39 9	0,977508	1,226707	2,8018	7,7184
1,9	0,984945	1,890200	108 18 2	0,950833	1,323040	3,0836	8,3247
2,0	0,983211	1,988609	113 56 20	0,914815	1,416285	3,3391	8,9113
2,1	0,981222	2,086833	119 34 0	0,869817	1,505544	3,5629	9,4724
2,2	0,978861	2,184841	125 10 56	0,816292	1,589966	3,7520	10,0025
2,3	0,975932	2,282586	130 46 57	0,754722	1,668753	3,9052	10,4965
2,4	0,972101	2,379997	136 21 50	0,685869	1,741171	4,0238	10,9494
2,5	0,966751	2,476956	141 55 9	0,610263	1,806562	4,1106	11,3570
2,6	0,958669	2,573257	147 26 12	0,528699	1,864351	4,1699	11,7151
2,7	0,945191	2,668514	152 53 41	0,441980	1,914072	4,2070	12,0202
2,8	0,919492	2,761901	158 14 43	0,350957	1,955396	4,2278	12,2696
2,9	0,859468	2,851328	163 22 9	0,256530	1,988213	4,2375	12,4605
3,0	0,656408	2,929514	167 50 56	0,159653	2,012904	4,2409	12,5914
3,1	-1,241484	2,942373	168 35 8	0,061500	2,032025	4,2427	12,6609

TABLE I

$\beta = -0,0150$

$S/b$	$b/\rho$	$\varphi$, radians	$\varphi$, degrees, min., sec.	$x/b$	$z/b$	$V/b^3$	$\Pi/b^2$
0,1	0,999946	0,099998	5 43 46	0,099833	0,004996	0,0001	0,0314
0,2	0,999775	0,199985	11 27 29	0,198670	0,019933	0,0012	0,1252
0,3	0,999499	0,299950	17 11 09	0,295521	0,044660	0,0061	0,2806
0,4	0,999109	0,399881	22 54 41	0,389422	0,078927	0,0189	0,4960
0,5	0,998615	0,499768	28 38 04	0,479437	0,122390	0,0450	0,7692
0,6	0,998017	0,599600	34 21 16	0,564671	0,174671	0,0900	1,0974
0,7	0,997317	0,699367	40 04 14	0,644277	0,235064	0,1599	1,4725
0,8	0,996524	0,799060	45 46 57	0,717469	0,303142	0,2593	1,9057
0,9	0,995636	0,898669	51 29 23	0,783525	0,378165	0,3927	2,3776
1,0	0,994662	0,998185	57 11 30	0,841796	0,459381	0,5616	2,8887
1,1	0,993603	1,097599	62 53 16	0,891714	0,545983	0,7665	3,4337
1,2	0,992464	1,196903	68 34 38	0,932797	0,637110	1,0051	4,0073
1,3	0,991242	1,296089	74 15 37	0,964647	0,731858	1,2739	4,6039
1,4	0,989943	1,395148	79 56 09	0,986965	0,829294	1,5656	5,2175
1,5	0,988560	1,494074	85 36 14	0,999544	0,928459	1,8734	5,8421
1,6	0,987086	1,592857	91 15 50	1,002274	1,028381	2,1885	6,4715
1,7	0,985508	1,691488	96 54 54	0,995144	1,128085	2,5017	7,0995
1,8	0,983809	1,789955	102 33 24	0,978239	1,226605	2,8029	7,7200
1,9	0,981945	1,888244	108 11 18	0,951738	1,322988	3,0858	8,3267
2,0	0,979867	1,986337	113 48 31	0,915917	1,416309	3,3420	8,9140
2,1	0,977487	2,084207	119 24 58	0,871137	1,505678	3,5667	9,4759
2,2	0,974665	2,181819	125 00 32	0,817850	1,590250	3,7566	10,0069
2,3	0,971169	2,279118	130 35 01	0,756586	1,669237	3,9109	10,5019
2,4	0,966605	2,376018	136 08 08	0,687952	1,741911	4,0305	10,9561
2,5	0,960246	2,472380	141 39 24	0,612626	1,807624	4,1184	11,3650
2,6	0,950673	2,567961	147 07 59	0,531347	1,865814	4,1785	11,7247
2,7	0,934777	2,662308	152 32 20	0,444913	1,916031	4,2165	12,0316
2,8	0,904660	2,754458	157 49 07	0,354172	1,957973	4,2378	12,2828
2,9	0,835063	2,841991	162 50 02	0,260021	1,991575	4,2481	12,4760
3,0	0,604775	2,916643	167 06 40	0,163416	2,017319	4,2517	12,6091
3,09	0,801931	2,931130	167 56 28	0,075363	2,035934	4,2505	12,6766

TABLE I

$\beta = -0,0250$

$\dfrac{s}{b}$	$\dfrac{b}{p}$	$\varphi$, radians	$\varphi$, deg. min. sec.	$\dfrac{x}{b}$	$\dfrac{z}{b}$	$\dfrac{v}{b^3}$	$\dfrac{n}{b^2}$
0,1	0,999907	0,099997	5 43 46	0,099833	0,004996	0,0001	0,0313
0,2	0,999627	0,199975	11 27 28	0,198670	0,019932	0,0012	0,1252
0,3	0,999161	0,299916	17 11 21	0,295522	0,044657	0,0061	0,2806
0,4	0,998514	0,399801	22 54 25	0,389425	0,078920	0,0190	0,4960
0,5	0,997690	0,499613	28 37 33	0,479445	0,122373	0,0451	0,7692
0,6	0,996693	0,599833	34 20 21	0,564689	0,174575	0,0902	1,0975
−0,7	0,995530	0,698946	40 2 48	0,644316	0,235001	0,1599	1,4776
0,8	0,994207	0,798434	45 44 49	0,717543	0,303042	0,2595	1,9058
0,9	0,992731	0,897782	51 26 21	0,783656	0,378015	0,3926	2,3728
1,0	0,991109	0,996975	57 7 21	0,842012	0,459171	0,5616	2,8889
1,1	0,989347	1,095999	62 47 46	0,892052	0,545703	0,7664	3,4341
1,2	0,987449	1,194840	68 27 33	0,933300	0,636755	1,0052	4,0080
1,3	0,985419	1,293485	74 6 40	0,965371	0,731430	1,2738	4,6050
1,4	0,983256	1,391919	79 45 4	0,987972	0,828801	1,5661	5,2194
1,5	0,980956	1,490131	85 22 42	1,000906	0,927920	1,8745	5,8444
1,6	0,978507	1,588106	90 59 30	1,004070	1,027830	2,1905	6,4748
1,7	0,975888	1,685827	96 35 27	0,997458	1,127571	2,5048	7,1041
1,8	0,973064	1,783276	102 10 27	0,981160	1,226194	2,8086	7,7262
1,9	0,969979	1,880431	107 44 27	0,955357	1,322767	3,0935	8,3351
2,0	0,966545	1,977261	113 17 19	0,920323	1,416387	3,3527	8,9248
2,1	0,962623	2,073724	118 48 56	0,876419	1,506190	3,5809	9,4897
2,2	0,957992	2,169762	124 19 55	0,824086	1,591359	3,7748	10,0243
2,3	0,952288	2,265287	129 47 29	0,763847	1,671131	3,9333	10,5236
2,4	0,944893	2,360164	135 13 39	0,696297	1,744816	4,0571	10,9827
2,5	0,934695	2,454173	140 36 50	0,622097	1,811801	4,1490	11,3972
2,6	0,919535	2,546940	145 55 44	0,541970	1,871572	4,2130	11,7632
2,7	0,894795	2,637767	151 7 58	0,456697	1,923740	4,2543	12,0772
2,8	0,849089	2,725220	156 8 37	0,367107	1,968096	4,2782	12,3362
2,9	0,747716	2,805816	160 45 41	0,274091	2,004736	4,2903	12,5378
3,0	0,438992	2,868448	164 21 0	0,178632	2,034473	4,2951	12,6801
3,1	−1,555122	2,849408	163 15 33	0,082199	2,060941	4,2952	12,7620

TABLE I

$\beta = -0,0375$

$\dfrac{S}{b}$	$\dfrac{\rho}{b}$	$\varphi$, radians	$\varphi$, deg min sec	$\dfrac{x}{b}$	$\dfrac{z}{b}$	$\dfrac{V}{b^3}$	$\dfrac{\Pi}{b^2}$
0,1	0,999860	0,099995	5 43 45	0,099833	0,004996	0,0001	0,0313
0,2	0,999440	0,199962	11 27 25	0,198670	0,019931	0,0012	0,1252
0,3	0,998742	0,299874	17 10 53	0,295523	0,044654	0,0061	0,2806
0,4	0,997772	0,399702	22 54 4	0,389428	0,078910	0,0190	0,4960
0,5	0,996535	0,499419	28 36 55	0,479454	0,122350	0,0451	0,7692
0,6	0,995041	0,599000	34 19 13	0,564712	0,174531	0,0901	1,0975
0,7	0,993297	0,698419	40 0 59	0,644365	0,234923	0,1598	1,4776
0,8	0,991315	0,797651	45 42 7	0,717637	0,302916	0,2593	1,9058
0,9	0,989103	0,896674	51 22 32	0,783820	0,377827	0,3924	2,3779
1,0	0,986673	0,995464	57 2 9	0,842282	0,458908	0,5613	2,8892
1,1	0,984034	1,094001	62 40 54	0,892473	0,545353	0,7661	3,4346
1,2	0,981193	1,192264	68 18 42	0,933929	0,636311	1,0049	4,0088
1,3	0,978154	1,290233	73 55 30	0,966275	0,730892	1,2737	4,6063
1,4	0,974918	1,387888	79 31 12	0,989230	0,828181	1,5664	5,2211
1,5	0,971478	1,485210	85 5 47	1,002606	0,927243	1,8756	5,8474
1,6	0,967818	1,582177	90 39 7	1,006311	1,027135	2,1927	6,4790
1,7	0,963908	1,678765	96 11 10	1,000346	1,126917	2,5088	7,1099
1,8	0,959696	1,774948	101 41 49	0,984806	1,225663	2,8149	7,7340
1,9	0,955104	1,870692	107 10 58	0,959875	1,322466	3,1029	8,3454
2,0	0,950005	1,965952	112 38 27	0,925827	1,416451	3,3553	8,9303
2,1	0,944203	2,060669	118 4 3	0,883017	1,506783	3,5385	9,5070
2,2	0,937387	2,154759	123 27 31	0,831882	1,592677	3,7274	10,0462
2,3	0,929051	2,248096	128 48 23	0,772933	1,673409	3,9612	10,5207
2,4	0,918342	2,340490	134 6 1	0,706748	1,748326	4,0595	11,0159
2,5	0,903749	2,431636	139 19 21	0,633974	1,816861	4,1877	11,4375
2,6	0,882394	2,521018	144 26 37	0,555314	1,878553	4,2566	11,8114
2,7	0,848271	2,607697	149 22 16	0,471527	1,933083	4,3021	12,1342
2,8	0,787107	2,689794	154 6 50	0,383429	1,980339	4,3295	12,4029
2,9	0,657831	2,762942	158 18 18	0,291304	2,020573	4,3442	12,6153
3,0	0,298992	2,814268	161 14 44	0,197980	2,054867	4,3507	12,7693
3,1	-1,471061	2,783296	159 28 16	0,103554	2,087196	4,3525	12,8639

TABLE I

\[ \beta=-0,0500 \]

$S/b$	$\rho/b$	$\varphi$, radians	$\varphi$, deg. min. sec.	$x/b$	$z/b$	$V/b^3$	$\Pi/b^2$
0,1	0,999813	0,099994	5 43 45	0,099833	0,004996	0,0001	0,0313
0,2	0,999252	0,199950	11 27 23	0,198670	0,019951	0,0012	0,1252
0,3	0,998322	0,299832	17 10 45	0,295523	0,044651	0,0061	0,2806
0,4	0,997028	0,399602	22 53 44	0,389431	0,078901	0,0190	0,4960
0,5	0,995380	0,499226	28 36 13	0,479464	0,122328	0,0451	0,7692
0,6	0,993388	0,598667	34 18 4	0,564735	0,174487	0,0901	1,0975
0,7	0,991065	0,697892	39 59 11	0,644413	0,234845	0,1598	1,4776
0,8	0,988424	0,796869	45 39 26	0,717729	0,302791	0,2593	1,9059
0,9	0,985479	0,895567	51 18 44	0,783982	0,377640	0,3923	2,3781
1,0	0,982244	0,993955	56 56 58	0,842551	0,458644	0,5612	2,8895
1,1	0,978729	1,092006	62 34 2	0,892892	0,545003	0,7660	3,4351
1,2	0,974948	1,189692	68 9 52	0,934555	0,635867	1,0048	4,0097
1,3	0,970907	1,286987	73 44 20	0,967176	0,730555	1,2737	4,6076
1,4	0,966602	1,383865	79 17 23	0,990484	0,827561	1,5668	5,2231
1,5	0,962029	1,480299	84 48 54	1,004301	0,926562	1,8767	5,8503
1,6	0,957167	1,576261	90 18 47	1,008546	1,026432	2,1950	6,4831
1,7	0,951968	1,671721	95 46 57	1,003227	1,126254	2,5138	7,1156
1,8	0,946393	1,766742	101 13 16	0,988444	1,225118	2,8213	7,7418
1,9	0,940316	1,860982	106 37 35	0,964385	1,322143	3,1124	8,3558
2,0	0,933585	1,954684	111 59 42	0,931322	1,416480	3,3793	8,9518
2,1	0,925955	2,047670	117 19 22	0,889610	1,507325	3,6162	9,5243
2,2	0,917035	2,139832	122 36 12	0,839676	1,593926	3,8201	10,0670
2,3	0,906201	2,231013	127 49 40	0,782024	1,675591	3,9893	10,5778
2,4	0,892408	2,320974	132 58 55	0,717218	1,751706	4,1240	11,0492
2,5	0,873828	2,409336	138 2 41	0,645889	1,821747	4,2268	11,4777
2,6	0,847052	2,495468	142 58 47	0,568722	1,885301	4,3009	11,8596
2,7	0,805130	2,578248	147 43 22	0,486461	1,942110	4,3511	12,1913
2,8	0,732128	2,655476	152 8 51	0,399906	1,992143	4,3824	12,4700
2,9	0,584548	2,722248	155 58 24	0,309944	2,035768	4,4009	12,6931
3,0	0,206100	2,764877	158 24 57	0,217649	2,074240	4,4172	12,8503
3,1	-1,330544	2,725841	156 10 45	0,124959	2,111649	4,4207	12,9665

TABLE I

\[ \beta=-0,0625 \]

$S/b$	$b/\rho$	$\varphi$, radians	$\varphi$, degrees, min., sec.	$x/b$	$z/b$	$V/b^3$	$P/b^2$
0,1	0,999766	0,099992	5 43 45	0,099833	0,004996	0,0001	0,0313
0,2	0,999065	0,199938	11 27 20	0,198670	0,019930	0,0012	0,1252
0,3	0,997903	0,299790	17 10 36	0,295524	0,044648	0,0061	0,2806
0,4	0,996286	0,399503	22 53 23	0,389434	0,078891	0,0190	0,4960
0,5	0,994227	0,499032	28 35 33	0,479473	0,122506	0,0451	0,7692
0,6	0,991739	0,598334	34 16 55	0,564758	0,174442	0,0901	1,0975
0,7	0,988836	0,697366	39 57 22	0,644463	0,234767	0,1597	1,4777
0,8	0,985538	0,796088	45 36 45	0,717323	0,302665	0,2592	1,9060
0,9	0,981860	0,894461	51 14 56	0,784146	0,377452	0,3922	2,3782
1,0	0,977821	0,992448	56 51 47	0,842820	0,458381	0,5609	2,8898
1,1	0,973436	1,090014	62 27 12	0,893312	0,544651	0,7656	3,4356
1,2	0,968718	1,187124	68 1 2	0,935182	0,635420	1,0045	4,0105
1,3	0,963676	1,283747	73 33 12	0,968076	0,729814	1,2736	4,6089
1,4	0,958310	1,379849	79 3 34	0,991735	0,826935	1,5670	5,2251
1,5	0,952611	1,475398	84 32 3	1,005994	0,925875	1,8777	5,8532
1,6	0,946554	1,570359	89 58 30	1,010777	1,025723	2,1971	6,4872
1,7	0,940095	1,664695	95 22 48	1,006102	1,125576	2,5167	7,1213
1,8	0,933156	1,758362	100 44 48	0,992074	1,224551	2,8275	7,7496
1,9	0,925616	1,851306	106 4 19	0,968887	1,321788	3,1217	8,3661
2,0	0,917287	1,943459	111 21 7	0,936809	1,416466	3,3922	8,9652
2,1	0,907877	2,034728	116 34 53	0,896196	1,507899	3,6337	9,5415
2,2	0,896932	2,124984	121 45 9	0,847469	1,595096	3,8426	10,0897
2,3	0,883726	2,214039	126 51 18	0,791120	1,677668	4,0172	10,6048
2,4	0,867060	2,301614	131 52 22	0,727705	1,754948	4,1577	11,0823
2,5	0,844868	2,387269	136 46 50	0,657839	1,826450	4,2659	11,5179
2,6	0,813361	2,470231	141 32 12	0,582191	1,891808	4,3451	11,9078
2,7	0,765007	2,549389	146 4 9	0,501489	1,950817	4,3999	12,2485
2,8	0,683104	2,622189	150 14 25	0,416524	2,003510	4,4351	12,5371
2,9	0,524415	2,683534	153 45 19	0,328184	2,050338	4,4557	12,7712
3,0	0,147423	2,720181	155 51 18	0,237572	2,092627	4,4665	12,9490
3,1	−1,135848	2,685771	153 53 0	0,146612	2,134165	4,4717	13,0696

TABLE I

\[ \beta = -0,0750 \]

$S/b$	$b/\rho$	$\varphi$, radians	$\varphi$, deg. min. sec.	$x/b$	$z/b$	$V/b^3$	$\Pi/b^2$
0,1	0,999720	0,099991	5 43 45	0,099833	0,004995	0,0001	0,0313
0,2	0,998879	0,199925	11 27 17	0,198670	0,019930	0,0012	0,1252
0,3	0,997483	0,299748	17 10 27	0,295525	0,044645	0,0062	0,2806
0,4	0,995544	0,399404	22 53 3	0,389437	0,078882	0,0190	0,4960
0,5	0,993073	0,498839	28 34 53	0,479483	0,122283	0,0451	0,7692
0,6	0,990089	0,598001	34 15 47	0,564781	0,174398	0,0901	1,0975
0,7	0,986609	0,696840	39 55 34	0,644511	0,234689	0,1597	1,4777
0,8	0,982654	0,795307	45 34 4	0,712916	0,302540	0,2591	1,9061
0,9	0,978246	0,893356	51 11 8	0,784309	0,377265	0,3920	2,3784
1,0	0,973405	0,990942	56 46 36	0,843088	0,458117	0,5607	2,8901
1,1	0,968151	1,088023	62 20 21	0,893731	0,544300	0,7653	3,4361
1,2	0,962500	1,184559	67 52 13	0,935807	0,634974	1,0042	4,0113
1,3	0,956463	1,280510	73 22 4	0,968974	0,729273	1,2734	4,6102
1,4	0,950040	1,375839	78 49 47	0,992984	0,826309	1,5673	5,2270
1,5	0,943221	1,470505	84 15 13	1,007682	0,925185	1,8786	5,8560
1,6	0,935980	1,564469	89 38 15	1,013002	1,025006	2,1993	6,4913
1,7	0,928263	1,657685	94 58 42	1,008971	1,124889	2,5205	7,1270
1,8	0,919984	1,750103	100 16 25	0,995698	1,223968	2,8357	7,7573
1,9	0,911003	1,841659	105 31 9	0,973380	1,321409	3,1309	8,3764
2,0	0,901106	1,932273	110 42 40	0,942290	1,416418	3,4052	8,9786
2,1	0,889965	2,021839	115 50 34	0,902777	1,508244	3,6512	9,5587
2,2	0,877068	2,110208	120 54 22	0,855259	1,596196	3,8652	10,1114
2,3	0,861607	2,197167	125 53 18	0,800222	1,679650	4,0453	10,6319
2,4	0,842264	2,282401	130 46 19	0,738209	1,758051	4,1914	11,1155
2,5	0,816794	2,365418	135 31 42	0,669822	1,830982	4,3054	11,5582
2,6	0,781155	2,445424	140 6 45	0,595717	1,898088	4,3902	11,9561
2,7	0,727505	2,521057	144 26 45	0,516609	1,959219	4,4499	12,3057
2,8	0,639009	2,589787	148 23 2	0,433278	2,014465	4,4893	12,6043
2,9	0,474165	2,646588	151 37 37	0,346614	2,064332	4,5133	12,8495
3,0	0,107538	2,678199	153 26 58	0,257748	2,110179	4,5266	13,0394
3,1	−0,983608	2,645630	151 35 0	0,168634	2,155545	4,5331	13,1733

TABLE 1

\[ \beta=-0.0875 \]

$S/b$	$b/\rho$	$\varphi$, radians	$\varphi$, deg min sec	$x/b$	$z/b$	$V/b^3$	$\Pi/b^2$
0,1	0,999673	0,099989	5 43 44	0,099833	0,004995	0,0001	0,0313
0,2	0,998692	0,199912	11 27 15	0,198670	0,019929	0,0012	0,1252
0,3	0,997064	0,299706	17 10 19	0,295526	0,044642	0,0061	0,2806
0,4	0,994802	0,399304	22 52 42	0,389441	0,078873	0,0190	0,4960
0,5	0,991921	0,498646	28 34 13	0,479492	0,122261	0,0451	0,7692
0,6	0,988440	0,597668	34 14 38	0,564804	0,174354	0,0901	1,0975
0,7	0,984383	0,696314	39 53 45	0,644560	0,234611	0,1597	1,4778
0,8	0,979773	0,794527	45 31 23	0,718008	0,302414	0,2590	1,9061
0,9	0,974636	0,892251	51 7 20	0,784472	0,377077	0,3918	2,3785
1,0	0,968997	0,989437	56 41 26	0,843355	0,457854	0,5604	2,8903
1,1	0,962876	1,086034	62 13 31	0,894149	0,543948	0,7651	3,4366
1,2	0,956295	1,181997	67 43 24	0,936430	0,634528	1,0040	4,0122
1,3	0,949267	1,277279	73 10 58	0,969870	0,728731	1,2733	4,6115
1,4	0,941792	1,371835	78 36 1	0,994231	0,825680	1,5676	5,2290
1,5	0,933861	1,465622	83 58 26	1,009367	0,924490	1,8796	5,8589
1,6	0,925444	1,558592	89 18 3	1,015223	1,024282	2,2014	6,4955
1,7	0,916481	1,650693	94 34 40	1,011833	1,124189	2,5243	7,1328
1,8	0,906876	1,741866	99 48 6	0,999314	1,223368	2,8398	7,7650
1,9	0,896475	1,832041	104 58 6	0,977565	1,321007	3,1401	8,3866
2,0	0,885042	1,921127	110 4 21	0,947761	1,416332	3,4182	8,9920
2,1	0,872214	2,009003	115 6 27	0,909350	1,508626	3,6686	9,5759
2,2	0,857435	2,095505	120 3 49	0,863047	1,597225	3,8877	10,1331
2,3	0,839831	2,180397	124 55 39	0,809327	1,681535	4,0733	10,6588
2,4	0,817990	2,263331	129 40 46	0,748726	1,761044	4,2253	11,1487
2,5	0,789545	2,343777	134 17 19	0,681834	1,835342	4,3451	11,5984
2,6	0,750306	2,420884	138 42 23	0,609296	1,904140	4,4356	12,0043
2,7	0,692327	2,493222	142 51 4	0,531811	1,967320	4,5005	12,3630
2,8	0,599094	2,558196	146 34 26	0,450154	2,025015	4,5445	12,6717
2,9	0,431725	2,610633	149 34 42	0,365216	2,077771	4,5722	12,9280
3,0	0,081006	2,638667	151 11 4	0,278143	2,126941	4,5884	13,1302
3,1	-0,852004	2,608644	149 27 51	0,190889	2,175789	4,5968	13,2774

TABLE I

\[ \beta=-0.1000 \]

$S/b$	$b/\rho$	$\varphi$, radians	$\varphi$, deg. min. sec.	$x/b$	$z/b$	$V/b^3$	$\Pi/b^2$
0.1	0.999626	0.099987	5 43 44	0.099833	0.004995	0.0001	0.0313
0.2	0.998505	0.199900	11 27 12	0.198670	0.019928	0.0012	0.1252
0.3	0.996645	0.299664	17 10 10	0.295526	0.044639	0.0062	0.2806
0.4	0.994060	0.399205	22 52 22	0.389444	0.078863	0.0190	0.4960
0.5	0.990768	0.498452	28 33 33	0.479501	0.122239	0.0451	0.7692
0.6	0.986793	0.597336	34 13 29	0.564827	0.174309	0.0900	1.0975
0.7	0.982159	0.695789	39 51 57	0.644609	0.234533	0.1596	1.4778
0.8	0.976895	0.793746	45 28 42	0.718101	0.302289	0.2589	1.9062
0.9	0.971030	0.891148	51 3 32	0.784635	0.376890	0.3917	2.3787
1.0	0.964593	0.987933	56 36 16	0.843623	0.457591	0.5601	2.8906
1.1	0.957610	1.084048	62 6 41	0.894566	0.543597	0.7648	3.4371
1.2	0.950104	1.179438	67 34 37	0.937052	0.634081	1.0037	4.0130
1.3	0.942088	1.274052	72 59 52	0.970764	0.728188	1.2732	4.6128
1.4	0.933568	1.367839	78 22 17	0.995474	0.825049	1.5678	5.2310
1.5	0.924531	1.460748	83 41 41	1.011047	0.923792	1.8806	5.8618
1.6	0.914945	1.552726	88 57 53	1.017439	1.023553	2.2035	6.4996
1.7	0.904747	1.643717	94 10 41	1.014689	1.123475	2.5281	7.1384
1.8	0.893832	1.733652	99 19 51	1.002922	1.222752	2.8459	7.7728
1.9	0.882032	1.822454	104 25 8	0.982342	1.320578	3.1492	8.3968
2.0	0.869092	1.910021	109 26 10	0.953225	1.416211	3.4310	9.0054
2.1	0.854623	1.996222	114 22 30	0.915918	1.508958	3.6859	9.5930
2.2	0.838030	2.080875	119 13 31	0.870831	1.598183	3.9101	10.1547
2.3	0.818385	2.163726	123 58 21	0.818435	1.683524	4.1013	10.6858
2.4	0.794217	2.244402	128 35 41	0.759257	1.763900	4.2591	11.1818
2.5	0.763066	2.322339	133 3 37	0.693875	1.839532	4.3852	11.6386
2.6	0.720692	2.396644	137 19 3	0.622923	1.909968	4.4814	12.0525
2.7	0.659206	2.465845	141 16 57	0.547092	1.975124	4.5517	12.4203
2.8	0.562714	2.527337	144 48 21	0.467146	2.035171	4.6006	12.7392
2.9	0.395372	2.576059	147 35 50	0.383979	2.090676	4.6328	13.0067
3.0	0.062948	2.600981	149 1 31	0.298743	2.142966	4.6537	13.2212
3.1	−0.742139	2.573188	147 25 58	0.213382	2.194946	4.6688	13.3820
3.2	−3.330443	2.394339	137 11 8	0.133136	2.254226	4.6972	13.4903

The values of $\varphi$, $x/b$, $z/b$, and $V/b^3$ coincide with those in the tables of Bamford and Adams, who computed them for $\beta=-0.1000$ only up to $S/b=2.0$.

TABLE I

\[ \beta=-0.1125 \]

$\dfrac{S}{b}$	$\dfrac{b}{\rho}$	$\varphi$, radians	$\varphi$, deg. min. sec.	$\dfrac{x}{b}$	$\dfrac{z}{b}$	$\dfrac{V}{b^3}$	$\dfrac{M}{b^2}$
0,1	0,999578	0,099986	5 43 44	0,099833	0,004995	0,0001	0,0313
0,2	0,998317	0,199888	11 27 10	0,198670	0,019928	0,0012	0,1252
0,3	0,996225	0,299622	17 10 2	0,295527	0,044636	0,0062	0,2806
0,4	0,993318	0,399106	22 52 2	0,389447	0,078854	0,0190	0,4960
0,5	0,989616	0,498259	28 32 53	0,479511	0,122217	0,0451	0,7692
0,6	0,985146	0,597003	34 12 21	0,564850	0,174265	0,0900	1,0975
0,7	0,979936	0,695264	39 50 8	0,644657	0,234455	0,1596	1,4778
0,8	0,974020	0,792967	45 26 1	0,718194	0,302163	0,2588	1,9063
0,9	0,967429	0,890045	50 59 45	0,784797	0,376702	0,3915	2,3789
1,0	0,960197	0,986432	56 31 6	0,843891	0,457327	0,5600	2,8909
1,1	0,952354	1,082064	61 59 52	0,894983	0,543245	0,7645	3,4376
1,2	0,943924	1,176883	67 25 50	0,937674	0,633633	1,0034	4,0138
1,3	0,934926	1,270830	72 48 48	0,971657	0,727642	1,2731	4,6141
1,4	0,925365	1,363849	78 8 34	0,996715	0,824414	1,5681	5,2330
1,5	0,915230	1,455884	83 24 58	1,012725	0,923088	1,8815	5,8647
1,6	0,904485	1,546875	88 37 46	1,019649	1,022814	2,2055	6,5037
1,7	0,893063	1,636759	93 46 46	1,017539	1,122758	2,5317	7,1441
1,8	0,880852	1,725462	98 51 42	1,006523	1,222116	2,8519	7,7804
1,9	0,867674	1,812897	103 52 17	0,986810	1,320121	3,1582	8,4071
2,0	0,853255	1,898955	108 48 8	0,958679	1,416051	3,4438	9,0187
2,1	0,837185	1,983493	113 38 45	0,922426	1,509235	3,7033	9,6101
2,2	0,818838	2,066317	118 23 28	0,878609	1,599068	3,9326	10,1763
2,3	0,797751	2,147154	123 1 22	0,827542	1,685014	4,1295	10,7127
2,4	0,770901	2,225609	127 31 5	0,769795	1,766624	4,2955	11,2148
2,5	0,737302	2,301094	131 50 35	0,705938	1,843559	4,4254	11,6787
2,6	0,692205	2,372690	135 56 42	0,636593	1,915571	4,5277	12,1008
2,7	0,627904	2,438902	139 44 20	0,562441	1,982636	4,6037	12,4777
2,8	0,529330	2,497146	143 4 33	0,484241	2,044942	4,6576	12,8067
2,9	0,368930	2,542586	145 40 46	0,402881	2,103069	4,6937	13,0855
3,0	0,051412	2,565169	146 58 24	0,319519	2,158299	4,7165	13,3125
3,1	−0,644572	2,540618	145 34 0	0,236039	2,213551	4,7299	13,4870
3,2	−2,572920	2,397363	137 21 32	0,156914	2,274367	4,7373	13,6100
3,3	−7,812570	1,906544	109 14 13	0,098890	2,354593	4,7415	13,6882

TABLE I

\[ \beta=-0{,}1250 \]

$s/b$	$\rho/b$	$\varphi$, radians	$\varphi$, deg. min. sec.	$x/b$	$z/b$	$V/b^3$	$\Pi/b^2$
0,1	0,999533	0,099984	5 43 43	0,099833	0,004995	0,0001	0,0313
0,2	0,998131	0,199875	11 27 7	0,198670	0,019927	0,0012	0,1252
0,3	0,995806	0,299580	17 9 53	0,295528	0,044633	0,0062	0,2806
0,4	0,992576	0,399006	22 51 41	0,389450	0,078844	0,0190	0,4960
0,5	0,988465	0,498066	28 32 13	0,479520	0,122194	0,0450	0,7692
0,6	0,983501	0,596671	34 11 12	0,564873	0,174221	0,0900	1,0976
0,7	0,977716	0,694738	39 48 20	0,644706	0,234377	0,1596	1,4779
0,8	0,971148	0,792188	45 23 20	0,718286	0,302058	0,2587	1,9064
0,9	0,963833	0,888943	50 55 58	0,784960	0,376515	0,3913	2,3790
1,0	0,955808	0,984931	56 25 57	0,844157	0,457063	0,5597	2,8912
1,1	0,947106	1,080082	61 53 3	0,895399	0,542892	0,7642	3,4381
1,2	0,937758	1,174331	67 17 3	0,938294	0,633184	1,0031	4,0146
1,3	0,927782	1,267613	72 37 44	0,972547	0,727096	1,2729	4,6154
1,4	0,917186	1,359866	77 54 52	0,997953	0,823778	1,5683	5,2349
1,5	0,905958	1,451029	83 8 16	1,014398	0,922381	1,8824	5,8676
1,6	0,894062	1,541036	88 17 41	1,021855	1,022069	2,2075	6,5077
1,7	0,881428	1,629817	93 22 53	1,020382	1,122025	2,5354	7,1498
1,8	0,867936	1,717293	98 23 37	1,010116	1,221465	2,8579	7,7881
1,9	0,853398	1,803370	103 19 32	0,991270	1,319641	3,1672	8,4173
2,0	0,837530	1,887929	108 10 13	0,964125	1,415855	3,4566	9,0320
2,1	0,819900	1,970817	112 55 10	0,929027	1,509463	3,7205	9,6272
2,2	0,799861	2,051829	117 33 40	0,886382	1,599884	3,9550	10,1979
2,3	0,776422	2,130677	122 4 44	0,836650	1,686611	4,1576	10,7395
2,4	0,748036	2,206949	126 26 56	0,780344	1,769223	4,3277	11,2478
2,5	0,712208	2,280038	130 38 12	0,718024	1,847401	4,4659	11,7189
2,6	0,664747	2,349007	134 35 17	0,650305	1,920955	4,5744	12,1490
2,7	0,598218	2,412358	138 13 5	0,577858	1,989861	4,6563	12,5351
2,8	0,498497	2,467559	141 22 51	0,501456	2,054336	4,7156	12,8743
2,9	0,336267	2,510012	143 48 47	0,421922	2,114967	4,7564	13,1645
3,0	0,043325	2,530563	144 59 26	0,340470	2,172979	4,7830	13,4041
3,1	−0,563761	2,508564	143 43 48	0,258912	2,230840	4,7994	13,5923
3,2	−2,056564	2,389496	136 54 29	0,181220	2,293710	4,8090	13,7302
3,3	−5,816097	2,019044	115 40 58	0,119849	2,371950	4,8142	13,8233
3,4	−7,030780	1,294624	74 10 35	0,110307	2,469204	4,8178	13,8918
3,5	−2,800439	0,814559	46 40 15	0,162325	2,554494	4,8221	13,9754

TABLE I

$\beta = -0,1500$

$s/b$	$\rho/b$	$\varphi$, radians	$\varphi$, deg. min. sec.	$x/b$	$z/b$	$\gamma/b^3$	$\eta/b^3$
0,1	0,999439	0,099981	5 43 43	0,099833	0,004995	0,0001	0,0313
0,2	0,997757	0,199850	11 27 2	0,198670	0,019926	0,0012	0,1252
0,3	0,994968	0,299496	17 9 35	0,295529	0,044627	0,0062	0,2806
0,4	0,991095	0,398808	22 51 0	0,389456	0,078826	0,0190	0,4960
0,5	0,986164	0,497679	28 30 54	0,479539	0,122150	0,0450	0,7692
0,6	0,980213	0,596006	34 8 55	0,564919	0,174132	0,0899	1,0976
0,7	0,973280	0,693689	39 44 44	0,644802	0,234221	0,1594	1,4779
0,8	0,965413	0,790631	45 17 59	0,718470	0,301787	0,2585	1,9067
0,9	0,956653	0,886741	50 48 23	0,785283	0,376140	0,3910	2,3793
1,0	0,947048	0,981935	56 15 38	0,844689	0,456535	0,5592	2,8918
1,1	0,936640	1,076124	61 39 27	0,896228	0,542188	0,7636	3,4391
1,2	0,925464	1,169235	66 59 32	0,939531	0,632286	1,0026	4,0163
1,3	0,913545	1,261192	72 15 40	0,974324	0,726000	1,2726	4,6180
1,4	0,900894	1,351320	77 27 34	1,000421	0,822499	1,5687	5,2388
1,5	0,887502	1,441346	82 34 59	1,017733	0,920956	1,8841	5,8733
1,6	0,873331	1,529395	87 37 40	1,026251	1,020559	2,2114	6,5159
1,7	0,858303	1,615984	92 35 21	1,026049	1,120528	2,5426	7,1611
1,8	0,842292	1,701023	97 27 41	1,017279	1,220112	2,8696	7,8034
1,9	0,825095	1,784403	102 14 20	1,000163	1,318608	3,1849	8,4377
2,0	0,806407	1,865992	106 54 49	0,974988	1,415359	3,4818	9,0586
2,1	0,785774	1,945620	111 28 33	0,942103	1,509769	3,7547	9,6612
2,2	0,762519	2,023060	115 54 46	0,901909	1,601308	3,9996	10,2409
2,3	0,735629	2,098003	120 12 24	0,854861	1,689523	4,2138	10,7931
2,4	0,703552	2,170013	124 19 57	0,801460	1,774044	4,3963	11,3138
2,5	0,663854	2,238461	128 15 16	0,742253	1,854607	4,5475	11,7991
2,6	0,612577	2,302401	131 55 4	0,677837	1,931078	4,6689	12,2454
2,7	0,542998	2,360371	135 14 22	0,608870	2,003467	4,7633	12,6499
2,8	0,443030	2,409998	138 4 58	0,536090	2,072032	4,8342	13,0098
2,9	0,289413	2,447208	140 12 53	0,460570	2,137340	4,8853	13,3229
3,0	0,033430	2,464518	141 12 23	0,382840	2,200498	4,9207	13,5879
3,1	−0,436884	2,446925	140 11 54	0,305208	2,263530	4,9442	13,8040
3,2	−1,399515	2,361152	135 17 3	0,230686	2,330168	4,9592	13,9720
3,3	−3,433795	2,131022	122 5 55	0,166998	2,406983	4,9686	14,0961
3,4	−5,826673	1,653055	94 42 47	0,133744	2,500269	4,9751	14,1882
3,5	−4,249255	1,116863	63 59 30	0,153375	2,597075	4,9810	14,2756

TABLE I

\[ \beta=-0{,}1750 \]

$S/b$	$b/\rho$	$\varphi$, radians	$\varphi$, deg. min. sec.	$x/b$	$z/b$	$V/b^3$	$\Pi/b^2$
0,1	0,999345	0,099978	5 43 42	0,099833	0,004995	0,0001	0,0313
0,2	0,997383	0,199825	11 26 57	0,198670	0,019925	0,0012	0,1252
0,3	0,994131	0,299411	17 9 18	0,295531	0,044620	0,0062	0,2806
0,4	0,989613	0,398609	22 50 19	0,389462	0,078807	0,0190	0,4960
0,5	0,983865	0,497293	28 29 34	0,479558	0,122105	0,0450	0,7692
0,6	0,976928	0,595342	34 6 38	0,564965	0,174044	0,0899	1,0976
0,7	0,968852	0,692640	39 41 7	0,644900	0,234095	0,1592	1,4780
0,8	0,959689	0,789076	45 12 39	0,718655	0,301536	0,2582	1,9067
0,9	0,949492	0,884543	50 40 50	0,785607	0,375764	0,3906	2,3797
1,0	0,938317	0,978942	56 5 21	0,845221	0,456006	0,5587	2,8924
1,1	0,926213	1,072175	61 25 52	0,897056	0,541481	0,7630	3,4401
1,2	0,913222	1,164154	66 42 5	0,940764	0,631384	1,0019	4,0179
1,3	0,899378	1,254791	71 53 39	0,976092	0,724898	1,2723	4,6206
1,4	0,884694	1,344002	77 0 20	1,002880	0,821209	1,5690	5,2427
1,5	0,869164	1,431702	82 1 50	1,021054	0,919511	1,8858	5,8790
1,6	0,852750	1,517805	86 57 50	1,030626	1,019021	2,2152	6,5240
1,7	0,835373	1,602219	91 48 1	1,031690	1,118985	2,5496	7,1723
1,8	0,816898	1,684843	96 32 4	1,024410	1,218691	2,8811	7,8187
1,9	0,797117	1,765555	101 9 32	1,009019	1,317472	3,2023	8,4579
2,0	0,775713	1,844212	105 39 56	0,985812	1,414716	3,5067	9,0850
2,1	0,752220	1,920628	110 2 38	0,955140	1,509870	3,7886	9,6951
2,2	0,725958	1,994564	114 16 48	0,917446	1,602453	4,0440	10,2838
2,3	0,695918	2,066694	118 21 20	0,873057	1,692058	4,2700	10,8466
2,4	0,660596	2,133572	122 14 41	0,822589	1,778366	4,4652	11,3796
2,5	0,617689	2,197562	125 54 40	0,766537	1,861159	4,6298	11,8791
2,6	0,563577	2,256738	129 18 6	0,705487	1,940342	4,7649	12,3418
2,7	0,492353	2,309711	132 20 12	0,640082	2,015972	4,8727	12,7647
2,8	0,393956	2,354311	134 53 32	0,571049	2,088310	4,9562	13,1453
2,9	0,250320	2,387011	136 45 56	0,499242	2,157901	5,0191	13,4817
3,0	0,027049	2,401757	137 36 38	0,425758	2,225724	5,0648	13,7723
3,1	-0,345299	2,387530	136 47 43	0,352173	2,293437	5,0970	14,0167
3,2	-1,010636	2,323079	133 6 9	0,281147	2,363609	5,1192	14,2154
3,3	-2,222832	2,167116	124 10 0	0,217990	2,441212	5,1342	14,3716
3,4	-3,951772	1,858875	106 30 20	0,174044	2,530597	5,1446	14,4933
3,5	-4,386140	1,420341	81 22 46	0,166842	2,629518	5,1531	14,5981

TABLE I

$S/b$	$b/\rho$	$\varphi$, radians	$\varphi$, deg. min. sec.	$x/b$	$z/b$	$V/b^3$	$\Pi/b^2$
0,1	0,999252	0,099975	5 43 41	0,099833	0,004995	0,0001	0,0313
0,2	0,997010	0,199800	11 26 52	0,198671	0,019923	0,0012	0,1252
0,3	0,993293	0,299327	17 9 1	0,295552	0,044614	0,0062	0,2806
0,4	0,988132	0,398411	22 49 38	0,389468	0,078788	0,0190	0,4960
0,5	0,981567	0,496907	28 28 14	0,479576	0,122061	0,0450	0,7692
0,6	0,973648	0,594679	34 4 21	0,565010	0,173956	0,0898	1,0976
0,7	0,964430	0,691593	39 37 51	0,644997	0,233903	0,1592	1,4781
0,8	0,953976	0,787523	45 7 18	0,718839	0,301286	0,2580	1,9069
0,9	0,942349	0,882349	50 33 18	0,785929	0,375389	0,3903	2,3800
1,0	0,929611	0,975956	55 55 5	0,845750	0,455477	0,5582	2,8920
1,1	0,915822	1,068237	61 12 19	0,897880	0,540775	0,7624	3,4411
1,2	0,901031	1,159086	66 24 39	0,941992	0,630480	1,0014	4,0196
1,3	0,885278	1,248410	71 31 43	0,977854	0,723793	1,2720	4,6231
1,4	0,868583	1,336111	76 33 13	1,005326	0,819911	1,5693	5,2466
1,5	0,850942	1,422095	81 28 48	1,024357	0,918054	1,8874	5,8847
1,6	0,832318	1,506266	86 18 10	1,034981	1,017457	2,2189	6,5321
1,7	0,812633	1,588523	91 0 56	1,037304	1,117402	2,5564	7,1835
1,8	0,791751	1,668753	95 36 45	1,031507	1,217206	2,8924	7,8339
1,9	0,769457	1,746826	100 5 9	1,017835	1,316241	3,2195	8,4781
2,0	0,745434	1,822586	104 25 35	0,996593	1,413936	3,5312	9,1113
2,1	0,719217	1,895839	108 37 25	0,968135	1,509779	3,8222	9,7289
2,2	0,690135	1,966534	112 39 46	0,932865	1,603328	4,0881	10,3265
2,3	0,657213	2,033739	116 31 29	0,891229	1,694227	4,3260	10,8998
2,4	0,619026	2,097602	120 11 2	0,843717	1,782200	4,5343	11,4452
2,5	0,573450	2,157298	123 36 15	0,790858	1,867070	4,7128	11,9589
2,6	0,517251	2,211939	126 44 6	0,733227	1,948778	4,8621	12,4379
2,7	0,445567	2,260229	129 30 6	0,671458	2,027407	4,9842	12,8795
2,8	0,349594	2,300222	131 47 35	0,606262	2,103224	5,0816	13,2810
2,9	0,216143	2,328903	133 26 11	0,538476	2,176738	5,1575	13,6407
3,0	0,020921	2,341412	134 9 11	0,469151	2,248808	5,2151	13,9573
3,1	-0,279462	2,329619	133 28 38	0,399738	2,320792	5,2578	14,2393
3,2	-0,762438	2,279488	130 36 18	0,332481	2,394782	5,2890	14,4601
3,3	-1,544884	2,167139	124 10 4	0,271266	2,473800	5,3115	14,6493
3,4	-2,658015	1,958857	112 14 3	0,223272	2,561318	5,3277	14,8037
3,5	-3,522396	1,642748	94 7 21	0,199848	2,658104	5,3408	14,9350
3,6	-3,135522	1,298737	74 24 44	0,210118	2,757074	5,3536	15,0620
3,7	-2,020212	1,040104	59 35 37	0,249946	2,848497	5,3685	15,2053
3,8	-1,105954	0,886876	50 48 51	0,307583	2,930098	5,3884	15,3798

The values of $\varphi$, $x/b$, $z/b$, and $V/b^3$ coincide with those in the tables of Bonford and Adams, who computed them for $\beta=-0,2000$ only up to $S/b=2,2$.

TABLE I

\[ \beta=-0{,}2250 \]

$s/b$	$b/\rho$	$\varphi$, radians	$\varphi$, deg	$\varphi$, min	$\varphi$, sec	$x/b$	$z/b$	$V/b^3$	$\Pi/b^2$
0,1	0,999138	0,099972	5	43	41	0,099833	0,004995	0,0001	0,0313
0,2	0,996636	0,199775	11	26	47	0,198671	0,019922	0,0012	0,1252
0,3	0,992455	0,299244	17	8	44	0,295534	0,044608	0,0062	0,2806
0,4	0,986652	0,398212	22	48	57	0,389475	0,078769	0,0190	0,4960
0,5	0,979271	0,496521	28	26	55	0,479595	0,122016	0,0449	0,7692
0,6	0,970371	0,594016	34	2	5	0,565056	0,173867	0,0898	1,0977
0,7	0,960016	0,690547	39	33	56	0,645093	0,233753	0,1591	1,4782
0,8	0,948275	0,785973	45	1	59	0,719023	0,301035	0,2578	1,9070
0,9	0,935223	0,880158	50	25	46	0,786251	0,375014	0,3899	2,3803
1,0	0,920931	0,972976	55	44	51	0,846278	0,454949	0,5577	2,8935
1,1	0,905468	1,064306	60	58	49	0,898701	0,540065	0,7618	3,4421
1,2	0,888891	1,154033	66	7	16	0,943214	0,629574	1,0007	4,0212
1,3	0,871247	1,242049	71	9	51	0,979607	0,722682	1,2715	4,6257
1,4	0,852561	1,328248	76	6	11	1,007762	0,818605	1,5696	5,2505
1,5	0,832836	1,412527	80	55	55	1,027646	0,916578	1,8889	5,8903
1,6	0,812035	1,494780	85	38	41	1,039313	1,015866	2,2224	6,5401
1,7	0,790084	1,574896	90	14	6	1,042889	1,115775	2,5631	7,1947
1,8	0,766847	1,652752	94	41	45	1,038569	1,215657	2,9034	7,8490
1,9	0,742110	1,728213	99	1	10	1,026611	1,314915	3,2363	8,4981
2,0	0,715562	1,801114	103	11	46	1,007328	1,413016	3,5554	9,1375
2,1	0,686746	1,871250	107	12	53	0,981082	1,509489	3,8554	9,7625
2,2	0,655015	1,938366	111	3	37	0,948279	1,603935	4,1319	10,3690
2,3	0,619446	2,002126	114	42	48	0,909369	1,696035	4,3819	10,9529
2,4	0,578718	2,062084	118	8	55	0,864834	1,785554	4,6035	11,5105
2,5	0,530911	2,117634	121	19	53	0,815199	1,872351	4,7962	12,0386
2,6	0,473200	2,167937	124	12	49	0,761033	1,956399	4,9605	12,5340
2,7	0,401328	2,211804	126	43	37	0,702961	2,037799	5,0978	12,9941
2,8	0,308735	2,247516	128	46	23	0,641681	2,116817	5,2102	13,4167
2,9	0,185059	2,272522	130	12	21	0,578010	2,193923	5,3004	13,7999
3,0	0,013585	2,282946	130	48	11	0,512951	2,269866	5,3715	14,1427
3,1	-0,232835	2,272758	130	13	10	0,447835	2,345760	5,4266	14,4445
3,2	-0,596679	2,232483	127	54	43	0,384583	2,423206	5,4688	14,7058
3,3	-1,133075	2,147649	123	3	4	0,326193	2,504352	5,5009	14,9288
3,4	-1,860687	1,999298	114	33	5	0,277514	2,591599	5,5253	15,1177
3,5	-2,586764	1,775095	101	42	20	0,245881	2,686243	5,5453	15,2810
3,6	-2,798781	1,499439	85	54	41	0,239079	2,785691	5,5633	15,4319
3,7	-2,294974	1,240237	71	3	37	0,259404	2,883316	5,5820	15,5872
3,8	-1,545476	1,048359	60	4	0	0,301282	2,973956	5,6042	15,7625

TABLE I

\[ \beta=-0.2500 \]

$S/b$	$b/\rho$	$\varphi$, radians	$\varphi$, deg. min. sec.	$x/b$	$z/b$	$V/b^3$	$\Pi/b^2$
0.1	0.999064	0.099969	5 43 40	0.099833	0.004995	0.0001	0.0313
0.2	0.996261	0.199751	11 26 41	0.198671	0.019921	0.0012	0.1252
0.3	0.991618	0.299160	17 8 26	0.295535	0.044602	0.0062	0.2806
0.4	0.985172	0.398014	22 48 16	0.389481	0.078750	0.0190	0.4960
0.5	0.976977	0.496136	28 25 35	0.479614	0.121972	0.0450	0.7692
0.6	0.967098	0.593353	33 59 48	0.565102	0.173778	0.0897	1.0977
0.7	0.955607	0.689502	39 30 20	0.645189	0.233597	0.1588	1.4782
0.8	0.942585	0.784424	44 56 39	0.719206	0.300784	0.2576	1.9072
0.9	0.928115	0.877970	50 18 14	0.786571	0.374639	0.3896	2.3806
1.0	0.912278	0.970001	55 34 37	0.846804	0.454419	0.5572	2.8941
1.1	0.895152	1.060383	60 45 20	0.899520	0.539356	0.7611	3.4431
1.2	0.876803	1.148991	65 49 56	0.944433	0.628666	1.0001	4.0228
1.3	0.857285	1.235705	70 48 2	0.981354	0.721567	1.2712	4.6282
1.4	0.836630	1.320410	75 39 14	1.010186	0.817289	1.5698	5.2543
1.5	0.814846	1.402995	80 23 8	1.030919	0.915086	1.8899	5.8960
1.6	0.791902	1.483340	84 59 21	1.043626	1.014249	2.2259	6.5481
1.7	0.767736	1.561335	89 27 28	1.048449	1.114106	2.5696	7.2058
1.8	0.742185	1.636840	93 47 2	1.045599	1.214042	2.9142	7.8640
1.9	0.715074	1.709717	97 57 34	1.035349	1.313493	3.2528	8.5181
2.0	0.686087	1.779798	101 58 29	1.018020	1.411960	3.5793	9.1636
2.1	0.654788	1.846857	105 49 2	0.993984	1.509038	3.8882	9.7960
2.2	0.620564	1.910653	109 28 20	0.963650	1.604279	4.1753	10.4113
2.3	0.582558	1.970845	112 55 16	0.927469	1.697488	4.4375	11.0057
2.4	0.539564	2.026998	116 8 18	0.885925	1.788435	4.6726	11.5757
2.5	0.489885	2.078535	119 5 29	0.839541	1.877015	4.8801	12.1180
2.6	0.431094	2.124673	121 44 5	0.788830	1.963222	5.0599	12.6297
2.7	0.359684	2.164355	124 0 26	0.734559	2.047174	5.2131	13.1085
2.8	0.270501	2.196020	125 49 22	0.677264	2.129128	5.3416	13.5522
2.9	0.155888	2.217592	127 3 31	0.617789	2.209517	5.4477	13.9591
3.0	0.004399	2.225974	127 32 20	0.557092	2.288988	5.5339	14.3282
3.1	−0.200841	2.216688	127 0 25	0.496395	2.368461	5.6032	14.6592
3.2	−0.483163	2.183242	125 5 26	0.437360	2.449170	5.6585	14.9524
3.3	−0.868444	2.116607	121 16 22	0.382372	2.532672	5.7025	15.2096
3.4	−1.362746	2.005833	114 55 33	0.334931	2.620645	5.7379	15.4344
3.5	−1.890151	1.842786	105 35 2	0.299919	2.714196	5.7669	15.6330
3.6	−2.229798	1.634040	93 37 25	0.282983	2.812566	5.7928	15.8151
3.7	−2.157619	1.410968	80 50 33	0.287861	2.912238	5.8179	15.9933
3.8	−1.739519	1.214297	69 34 27	0.313692	3.008676	5.8451	16.1813

The values of $\varphi$, $x/b$, and $z/b$ are more accurate than those given in Fordham’s tables and are computed only up to $S/b=3.2$.

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[v1] 1965

Results of the Numerical Solution of the Laplace Equation for Solids of Revolution with Shapes from - 0.0025 to - 0.2500

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RESULTS OF THE NUMERICAL SOLUTION OF THE LAPLACE EQUATION FOR BODIES OF REVOLUTION WITH SHAPES FROM −0.0025 TO −0.2500

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TABLE I

TABLE 1

TABLE I

TABLE I

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Results of the Numerical Solution of the Laplace Equation for Solids of Revolution with Shapes from - 0.0025 to - 0.2500