Tlte Variation in Attraction Due to the Attracting BodieS: 211 



When angle 3 becomes infinitesimal points d and f coincide with point 

 C, ard angle d A„g equals C A„g, and f b„g equals C b„g. Therefore infini- 

 tesimal angles C A„g: C b,,g :: B,*: A,'- 



Form Art. 10, 



A,2 = A'^- (A'2-B'2) sin^ ^. 

 B;- = B-^+ (A--B'-') sin-' 5. 

 Then A;-' + B,'^ = A'^ + B^ 



Angle C A„g represents any of the angles a C a,, a, C a^ etc., and C b„g^ 

 any of the corresponding angles b C b„ b, C bg, etc., of Dia. 5. The vol- 

 ume of infinitesimal wedge then from an oblate ellipsoid measured by A X 



a;- X — %„^o '■ equals that measured by A X B,- x Ang. ■' 



As A;^ + B;^=A'- + B-, the iufinitesimal wedges of prolate ellipsoids, the 

 sums of which make up the oblate ellipsoid are equal, each to each, in 

 volume. Ttie same can likewise be prpved of wedges of a prolate 

 ellipsoid. 



15. To find, for any even power, an expression for the average of the ec- 

 centricities of all the infinitesimal wedges composing an ellipsoid, with 

 edges in an equatorial diameter. 



The edges of the elliptic wedges have a common semi-axis A and 

 the other semi-axis varies from A to B as A, or B, vary from A to B. 



B, for angle 3- equals A, for angle (90° — 3). 



Eccentricity •^'~^''=:^— "— sin"5=E'^sin"3=E'^cos''(90° - 5). 



A- A^ 



Eccentricity •^—^=-^:^?!cos=3=E'cos''3=E'siQ'(90° - 3). 



A- A^ 



A=— A/^A^— B;^A^— A§^A^— B2 A=-A3 A^— Bi , „ , ,-o 

 ZT^ + rT-^+ r^r^+ rv^ + ;r-- + ^+&C. to 45° 



A-" A' A' A' A' A' 



Divided by the number of wedges equals, 

 E'(sin^5-f-cos'3-l-sin'S,-f-cos'-S-,-i-sin23,-i-cos-3g+&c to 45°), 

 divided by the number of wedges. 



Trigonometry gives: 



sin- ^ = i — i cos 2 5. 



cos- 5 = i 4- -I cos 2 3-. 



sin'' 5 + cos^ 3 

 ■ ^ = ■!• = sin" 45°= cos ^ 45.° 



The value i is true for any pair of wedges. It is therefore the true value 

 for the coefficient of E" in the expression E" sin'- 3 which gives i E'- for the 

 average eccentricity squared of all wedges. 



^4 -I- X4 + ^*c., to 3 for 45/ divided by number of wedges, 



equals E' ( sin' 3 + cos' ^) + etc., to 3 for 45,° divided by number of 

 wedsres. 



