256 Mr. J. Gordon on Dr. Lardner's and M. Lacroix's 



du, du* dy , dw, dw t dy 

 But, r- 1 - = j- 1 - . -^ and -^ = r-L . -y^- 

 dx dy dx dz dy dz 



d u l d z/! d y dy 



hence ^r;r = / (y) ~r= 



Case 2nd, when n = 2. 



T r* & u \ sf \ du^ s, \ du, dy 



By Case 1st, -jJ- =/(y)-^ =/(y)-^- - -|f ; 



where y*(j/) , ^ is a function of j/, and we may regard it 



therefore as the differential coefficient of a new function of y 



d u* d iio 







o ~ , x 

 which we represent by w 2 , so that . =/ (y) -~- 9 .. -- 



duc> dy .,, , fl N du, dy ... du, 

 bemg =-^--^ wdl be =/(y) ^- . -g = f(y) _J.. 



Consequently . 1 = . 2 ; and differentiating, , 8 ' = 



, dw, 



<P2 ^ W 2 do: , ^ 1 /- , , ^ W 2 



j -- ^~ ^ "^ ^ = j - 5 but by Case 1st -- = 

 dx . dz dz dx d % dx 



f(y) _fL, and we proved that -- =f(y) --, therefore, 



by substitution, 



3rd Case. In general if ^J = - ^^ 



du 



For a similar reason to that given in Case 2nd, for in- 

 troducing w 2 , we may suppose zv-i such a function of y, 



that *=* = /(5f)- -j; but by hypothesis 



du 



^ 7 . therefore by substitution = 



and, 



