Ckofton — Applications of the Method of Operative Symbols. 601 
To prove this, we have in general^ 
f{B)x = xf{D)+f'[D)', 
.-. B^x - xB^ = rB'^-'^ ; 
• *• ~; -~ — : = = . (ft. 
Hence it is easy to see that 
cix- xa = a(i>; 
.'. a{x - (}){x)) = xa= (z — ({>{z))a. (5) 
Hence (by writing x-<^{x) after each side of (5), &c.) it is easy to 
show for any power, positive or negative, oi x-<fi {x)^ 
Q.{x- <pxy = {z- (pzya, 
we conclude that for any function^ 
af{x-<l>x)=f{z-<})z)a; 
or, as any function of x can be put in the form f{x- <f}x), 
aF{x) = F{z)a. 
If, now, 1 be taken as operand^ 
aF{x)\ = F{z)ai. 
D2<^2 B^(}}^ 
Now, cLi= (^i+Bci>+-^+-^+ — y 
= 1+.^,+— + -^-^+.... 
= 1 + a<p'{x) = 1 + <t>'{z) ni ; 
1 dz 
thus equation (3), i.e. Lagrange's theorem, is proved. 
^ An operand will be understood in all cases, unless otherwise stated. 
2 First for any rational integral function ; any other function can he expressed 
as a series of rational integral functions. 
' A vertical bar indicates that the operations terminate there. 
2X2 
