4^ 



focus of the focal system being called the centre of the spheroid. The 

 general equation of such a spheroid contains four arbitrary constants, 

 of which three are the coordinates of the centre ; and if we determine 

 these four constants, by the condition that for some given values of 

 .r, y, z, that is for some given point of a given system, certain first 

 terms of the development 



V + 5F' + IJiF' + &c. 



may be equal to the corresponding terms of the development 



F + SF+iS^F + &c., 



the spheroid thus determined will be an osculating spheroid, to the 

 surface of constant action which passes through the given point of 

 the system. The conditions 



V'=z V, 3F' = 3F, (W") 



may be satisfied independently of the ratios of ix, hy, ^z, by taking 

 the centre of the spheroid any where upon the given ray, that is, by 

 establishing between the three coordinates of this centre the two 

 equations of the ray, and by assigning a proper value to the other 

 arbitrary constant ; there still remains therefore, after satisfying the 

 conditions (W"), an arbitrary parameter depending on the position of 

 the centre, which we may deteimine by the equation, 



J«F' = >*F, (X") 



assigning any arbitrary ratios to the three variations ix, iy, ^z, or 

 rather any value to the one ratio 



ydx — aiz 

 'Ay — /3^3 ' 



because, by the relations (H), '"''*^ ** " " 



