106 CONVERTIBILITY OF 



comparatively unimportant what that expression is, but it is 

 of some interest to notice the analogy of the result to those 

 in Arts. 127 and 128. 



Proofs of the proposition in the preceding Article have 

 sometimes been given which appear simpler than that here 

 adopted, but they are deficient in strictness. In particular 

 an assumption has sometimes been made which deserves to 

 be noticed. The following is substantially a proof that has 



jdlJb 

 fjlnf* 



been given. To obtain ? involves, according to the defi- 

 nition of the symbol, the following operations. (1) In the 

 function u we put x + h for x, subtract the original value 

 from the new value, and then divide by h. (2) We find the 

 limit of the result when h = 0. (3) We now put y + k for y, 

 subtract the original value from the new value, and then 

 divide by k. (4) We find the limit of the result when k = 0. 

 All this is immediately derived from first principles ; the 

 next step however is the assumption that we may perform 

 the third of the above operations before the second instead of 

 after it. With this assumption the required result is readily 

 obtained ; for from the first operation we get 



< (x + h, y) (f> (x, y) 



~k~ 

 then from the third we get 



(a? + h, y + k) - <f> (x + h, y) - $ (x, y + k) + <j> (ac, y} 

 hk 



,du 



fj/Y] 



and according to our assumption, the limit of this is . 



,du 



dy 

 And by a similar assumption it is found that -r^ is also 



equal to the same limit. 



One more remark must be made to guard against a possible 

 error. In the proof of Art. 134 we have used v for <"(# +6h, y} ; 

 in this expression all that is known of 6 is that it is a 

 proper fraction, and it must not be assumed to be a function 



