166 EVALUATION OF DIFFERENTIAL COEFFICIENTS 



. ... . fd'u\ ( tfu \ 

 In this expression we must substitute m [j-^j , I 3 / J > 



and [T-T] , the values of x and y under consideration, and thus 



di] 

 we obtain a quadratic for finding ~ . This quadratic is 



u\ = , , 



equation (2) agrees with equation (3) of Art. 181, remem- 

 bering that by hypothesis f -j- j = 0. 



191. Should the values of x and y we are considering in 



addition to making u = 0, ( ~j- } = 0, ( , J = 0, also make 



\dxj \dyj 



u \ .IIP dy 



r => tlien the value of f 



-7 r > 



\dxdyj dx 



given in equation (1) of the preceding Article also takes the 

 form - . Hence, applying again the rule for finding the 

 limit of such a fraction, we have 



(^] + 2 ( d * u \ dy + ( d * U Y^Yl ( ^ U \^ y 

 dy _ \dx*J \dx*dy)dx \dxdy* J\dx) \dxdy)dx* .. 



dx~~ / d s u \ / 'd*u \dy (d*u\(dy^ (d*u\<Fy"' ( >' 

 \dx 2 dyJ ' \dxdtf) dx "*" \dtf) \dx) * \dtf) dx* 



Since ( -j r } and (--.,) vanish, we obtain from (1) 

 \dxdyJ \dy J 



*?\ (*\* + 3 / <? \ ( d y\\ Q ( d * u \ ty + ( d * u \-o (^ 



dy 3 ) (dx) H 6 (dxdy*) (dx) 4 3 (dx*dy) dx + (dx s )~ 



where in all the differential coefficients of u we must sub- 

 stitute the values of x and y under consideration, giving a 



cubic equation to determine -^ . Compare Art. 184. 



