OF THE INDEPENDENT VARIABLE. 187 



By addition we have 



cT it tfu d?u _ d?u 1 d?u 1 d*u 2 du cot 6 du, 

 ^ + d^ + ^~d^ + f ! d^ + r'shi^d^ i + rdr If ~d6' 



208. The following example for two independent variables 

 is analogous to that in Art. 200 for one independent variable. 



If x =e 6 and y = eii is required to change the independent 

 variables from x and y to 6 and < in the expression 



d n u n _! d n u n(n-l] n _ 2 2 d n u 

 *~ n + '' ~^ + ~~ + '~ 



Let this expression be denoted by v n , and let v n+l denote 

 what it becomes when n is changed into n + 1 ; we shall 

 prove that 



dv dv 



For dv_dv n dx_ dv n 



dd~ dx de~ fa' 



dv n dv n dy dv n 

 and -=-= V -j-r =y -r 5 



d<p dy d<p * dy 



Now take any term in the expression represented by v n and 

 perform the following operations : differentiate the term with 

 respect to x and afterwards multiply by x ; differentiate the 

 term with respect to y and afterwards multiply by y ; then 

 add the two results. Take for example the (r + l) th term 

 which is 



and by performing the operations we obtain 



(-!) ...(n-r + l) f d*u 



' * 



\r 



d n 



n-rj- 



dy 



