8] on lagrange’s theorem. 41 
For instance, u representing a variable contained in the function a A, and taking 
a particular form of 0 Ui < ^\, 
(*)*»-«©».) m . 
From this it is easy to demonstrate 
¿{(£) iaH7i hKi) , ‘ Sli3/i1 (8>. 
«{(a)**s*}-iCaO*“i*s'*j 0), 
where a'h denotes -yr HA, as usual. Hence also 
ah 
(l0) - 
of which a particular case is 
a {(ffi)*“' 1 e¥u } - 3 {(a)*®" ^'“^1 ( n > : 
Also, (j^ h **~ 1 (F , u(P u ) = Fu for A = 0 (12). 
Hence the form in question for Fx is 
Fx ={ff ,r ' (iW/ “) ( 13 >; 
from which, differentiating with respect to u, and writing F instead of F\ 
Fx (d \h%v ... 
T^S = U) " (WX 
a well-known form of Lagrange’s theorem, almost equally important with the more 
usual one. It is easy to deduce (13) from (14). To do this, we have only to form 
the equation 
(“)• 
deduced from (14) by writing Fxf'x for fx, and adding this to (14), 
Fx — (^¡~) dh (Fu e h f u ) — A {^~ j dh (Am f'u e h f u ) 
= dk |jw el> ^ ~ ^ f ' U ehfU ^ 
“(k)**” (*"»**) (MX 
c. 6