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XLYII. 
APPLICATIONS OP THE METHOD OP OPERATIYE 
SYMBOLS. By M. W. CROPTOIN-, P. R. S. 
(communicated by JOHN CASEY, LL.D., F.E.S.) 
[Read November 10, 1890.] 
A FEW applications of symbolical methods to Lagrange's expansion, 
and one or two other questions in the Differential Calculus, are given 
in this short Paper. 
1. Lagrange's theorem is :~If 2 depends on x by the relation^ 
z = a; + ^(z), (1) 
F{z) = F{x) + <!> {X) F'{x) + + ^^^^3 + &c. 
A more symmetrical form is, putting (f){x) = <j>, 
or if we put O for the operator 
F{z) = J)-^ClDF{x); (2) 
or differentiating both sides 
F'{z)^ = ClF'{x); 
or putting F for F'y for any function F we shall have 
dz 
nF{x)=F{z)-, (3) 
we shall find this is a case of the following ; — If any operand be 
understood on both sides 
QF{x) =F{z)a. <4) 
1 No generality is gained by writing 3/ before (p (z), as usually done. 
