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



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

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

 case of the circle. 



Equation ( 7 ) is therefore of the general form 



Ax"" + By^ + (7^2 = 2^2^ 



1 + sm a 



in which 



sm a 



and may be written 



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

 a = jRl/l + sin a, b = By c^BVI — sin a. 



Its volume is 



4 4 



F= ^ irahc = - ttB^ cos cl 

 6 6 



which is the volume of the sphere multiplied bj cos «. 



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 

 or 



52 (a;'2 4. ^'2) _|_ ^2^3 = a2 62. (9) 



Transforming by means of the equations 

 a;' = 03^ + «i tan a 



y' = yi 



z' — z, sec a 



