202 ON THE CONSTANT TERM IN THE [25 



Also 



flc dc 



= -vy-jr (e, 2 e 2 ) + ,-/-;-..- . y" + terms of higher dimensions in e* e* and y 2 . 

 a (e 2 ) v a (y ) 



Hence, we have ultimately, when e, 2 = e 2 , and y 2 = 0, 



cZP dc 



T . y 2 rf (e 2 ) (I (e 2 ) 



Limit of ^ = ^ = -^. 



<*(/) 



in which y 2 is to be put = after the differentiations. The relation thus 

 deduced holds good for all values of e~. By equating the coefficients of e 1 

 on the two sides of the equation 



dP dc dP dc 



we find -f, -& , as before. 



jT A 



Also, by equating the coefficients of higher powers of e 2 , we obtain 

 other relations between the coefficients of terms of higher orders in the 

 value of P. 



Similarly, taking e\ y 2 , and yf to be related as in Case IV., we have, 

 by the same reasoning as before, 



dP dP , 



0= ,-, > . e ~ + ~T7-2\ (yr y") + terms of higher dimensions in e 2 and y? y 2 . 

 a (e-) a \y~) 



Also 

 O _ y-^. . e 2 + , , , (y^ y') + terms of higher dimensions in e" and y, 2 y 2 . 



Hence, we have ultimately, when e 2 = and y* = y 2 , 



dP dg 



T . . c y 2 y, 2 d(e?) die 2 ) 

 Limit of - r^ = TTT- = \ ' , 

 e- dP^ dg 



d (7) 



in which e" is to be put =0 after the differentiations. The result thus 

 deduced holds good for all values of y 2 . By equating the coefficients of y 2 



