286 On the Developme7it of Functions of the Form ^{z-^-x). 

 for 



dm fjm ^n+ 1 



■vf/z^n and -y — \|/^ 



dz"' ^ dz'" n + 1 



are of the following forms respectively : 



d'n] m(in 1 ') 



-J— 4/^. i{« = rI/'«-. /''-■;«. ?^.^J/'"-'s.^«-l+ \ ^ ' n(n-\) 

 uz 1.2 



and 



d 

 dz' 



^--1 

 dz- ^' 



mim—l), X . „ , n 

 + — y-^ (« + 1 ) "• • ^^^"'~^ . .r"- ' - &c. ; 



and it is clear that the former, integrated between the limits 

 relative to t, is equivalent to the latter. 



Applying this property successively to the terms of the de- 

 velopment, we have 



Now we have assumed -^ = — 1, 

 dz 



.'. x= —z-\-C 



Hitherto we have not performed any operation relative to 

 any variables but x and ^^ If therefore ,«' be assumed a func- 

 tion [a(pu) of u, the truth of the development is not affected. 

 C in this case must be a function of?/, let it be (/«). Then 



fu = z-\-x = z-\-a(pu, 

 or u=Y ^{z + afii) ', 



antl by the above theorem, 



F« = FF..~+ ^(FF,.)^«.«+|;(^FF,= .(H^)^ +&c., 



which is Laplace's theorem. 



Lagrange's theorem is obtained by supposing Q — u, in 

 which case 



x=.a[<^u)=^~z+u, or u=:iZ-\-a{^^u); 



and applying the theorem, we get 



Vu='E!s + Wzm.a+^{¥'z{<^uf)-^ +&c. 



r ^g \ ^^ ' ' 1.2 



