0/ Taylor's Theorem. 139 



aro absolutely identical, — a theorem which of course admits, 

 but not without a somewhat complicated process, of an a 'pos- 

 terio7'i direct proof; so that the operator 12 is to all intents 

 and purposes what Professor Cayley calls a semi-invariant 

 or pene-invariant, but to which I am accustomed to give the 

 name of a differentiant to <^x. 



Finally, it may be observed that a development for/i may 

 be obtained by the use of the ordinary Taylor's theorem for 

 several variables. If we make use of this method, and write 

 in addition to 



do dc d .d 



d .d d .e d .J ' 



&c. = &c., 



we shall obtain the noteworthy symbolical and absolute iden- 

 tity 



which may be verified, but not without some little trouble, by 

 direct expansion. 



If we use fl ! to signify that fl is to be used as a pure ope- 

 rator on the matter coming after it (operating that is to say 



* If we write 



A = ( - i2 . + Q,)h -\-Q^K' i-il.Ji^ + . . . , 



we ought to have e^ — '[ = 0, and the coefficients in the expansion of e-^ — 1 

 according to ascending powers of h ought all to vanish identically ; and so 

 they will be found to do, provided that in each such coefficient expressed 

 as the sum of the product of powers of 12^, i2,, i2^, . . . and of O , the 

 power of the dotted Q, be taken last in order. As soon as that expansion 

 18 made (but of course not before) we may write i2 . —Q = 0, and we may 

 readily calculate a priori the value of each power of (12 , — Q,) ; thus we 

 shall obtain 



(Q. -£2)- = i2.^— 2i20.4-i2'-^ = fi.^-20HQ'^ = 2i2i; 

 and so by a similar calculation, having first determined Q. .'-, Q. .^, i2 .'*, 

 &c., we shall obtain 



(i2.-i2)=' = 6r2„ (i2.-i2)4 = 24Q, + 12Qj^ &c. ; 



on substituting these values in A + j— ^ + -. ^ 3 + • • • *^^ coefficients ot 



the several powers of h will be found to vanish. 



The appearance in the above process of a zero whose powers are not 

 zero is a phenomenon which will not shock those who are acquainted with 

 Professor Peirce's discussions of possible algebras ; but it is new to find it 

 occur in working out a symbolical identity. 



