48 Trans. Acad. Sci. of St. Louis. 



It will be seen from equation ( 7 ) that the coefficients of 

 x 2 and z 2 are the same as the coefficients of x 2 and y 2 in the 

 case of the circle. 



Equation ( 7 ) is therefore of the general form 

 Ax 2 + By 2 + Cz 2 = R 2 , 



1 + sin a 



B = l 



1 



in which 



C = 



1 — sin a 



and may be written 



R 2 (l + sin a) + R 2 + R 2 (1 — sin a) = lf ^ 

 which is the equation to the ellipsoid, whose semi-axes are 

 a = RV \ + sin a, b = i?, c = RV\ — sin a. 



Its volume is 



4 4 



V= ~ frabc — = ttR s cos a 

 o 6 



which is the volume of the sphere multiplied by cos a. 



Lastly, let us apply this method of transformation to the 

 prolate spheroid, taking the longest axis as the y axis and 

 the equal axes as the x and z axes. 



Adapting its equation (Fig. 3), we have 



x' 2 + z' 2 y' 2 _ 

 a 2 "*■ b 2 ~ 

 or 



b 2 (x' 2 + z' 2 ) + a 2 y' 2 = a 2 b 2 . (9) 



Transforming by means of the equations 



x' = x 1 + ^ tan a 



y' = y, 



z' = z 1 sec a 



