118 EXAMPLES OF 



If then we put z =- in the result of the first Example, we 

 deduce 



restore yx for y and divide by x ; then 



4. If y = TTT- JT, expand y* in ascending powers 



of x. 



we have y V(l # 2 ) = # y ; 



therefore y* (1 x 2 ) = a? 2xy + y* ................ (1), 



x if 

 and y =-+2- X . 



v* 

 We must then put y = z + y x, 



A 



v 2 

 so that (y) = y - , and / (y) = y n . 



Thus y n = z n + x | S +1 + ... +,y | r ^i C^ 2 ^ 1 ) + .-(2), 



and after the differentiations are performed, we must put 



x 



- for z. 



The quadratic equation (1) which we have employed gives 



sv 



two values for y, namely - -- jj- - ^- ; the series which we 

 have obtained in (2) applies to the value with the upper sign. 





_ 

 this be expanded in ascending powers of x the first term is 



