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 — sin a 



in which 



and may be written 



/v2 -w2 »2 



+ D2+D2T1 ^TT\ = 1 ' ( 8 > 



i* 8 (l + sin«) i2 2 i2 2 (l — sina) 

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



a = RV\ + sin a, b = R, c = RV / l — sin«. 



Its volume is 



4 4 



V — w 7ra6c = 5 7r.fl 8 cos a 

 o o 



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 i/ 2 _ 



or 



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



Transforming by means of the equations 

 x' = as, + z 1 tan a 



. rf = Vi 



z' = z l sec a 



