60 LEIBNITZ'S THEOREM. 



Similarly 



d a u d 3 z dy d?z _ dy d*z n d?y dz d*y dz d?u 

 _ _ y __ I __ y. _ _i_ 2 - - __ (- 2 _ ___ I __ - - __ I __ z 

 dx 3 dx 3 dx do? dx dx 3 dx' dx dx* dx dx 3 



d*z dytfz^ tfy^dz d*y 

 dx 3 dx dx* dx* dx dx 3 



So far, then, as we have proceeded, the numerical coeffi- 

 cients follow the same law as those of the Binomial Theorem. 

 We may prove by the method of Induction that such will 

 always be the case. For assume 



d?u d n z dyd n ~ l z n (n - 1) d 



_ ., ___ 1_ /M _JL _ _L ^ _ >_ _ 



-^ 1.2 d 



n (n - 1) . . . (n - r + 1) d r y d n ~*z 

 ~~~ ~' r ^ 



lr 



Differentiate both sides with respect to x : then 

 d n+1 u d n "z dd 



n(n-l)... (n-r _ 



[r_ \dx r dx n ~ 



n (n - 1) ... (n - r) frf^y dr*z 



~ "-" 



d n ydz 

 + ...... + dx*dx 



Rearranging the terms, we have 



(n+ 1) n ... (n + 1 - r) d^ 



