40 
[8 
8. 
ON LAGRANGE’S THEOREM. 
[From the Cambridge Mathematical Journal, vol. hi. (1843), pp. 283—286.] 
The value given by Lagrange’s theorem for the expansion of any function of the 
quantity x, determined by the equation 
x = u + hfx (1), 
admits of being expressed in rather a remarkable symbolical form. The cl 'priori 
deduction of this, independently of any expansion, presents some difficulties; I shall 
therefore content myself with showing that the form in question satisfies the equations 
(2>. 
Fx = Fu for h = 0 (3), 
deduced from the equation (1), and which are sufficient to determine the expansion of 
Fx, considered as a function of u and h in powers of h. 
Consider generally the symbolical expression 
^ ( A rf*) =* 
(4), 
involving 
in general symbols of operation relative to any of 
the other variables 
entering into Eh. 
Then, if Eh be expansible in the form 
it is obvious that 
Eh-t (Ah m ) 
</> (h Jr) Eh - £ [<f>m . = £ {(<f>m . A) h m ] 
(6).