138 Prof. J. J. Sylvester on a Generalization 



and consequently 



and the first part of the theorem is demonstrated. It will of 

 course be understood that (fl . )* means not (H') . , but 

 n.nM. (to /■ factors). 



Lagrange's or any other rule for the Remainder in the old 

 Taylor's theorem may be extended to this generalization of it ; 

 that is to say, if in the development of /i we stop at the nth 

 term, the remainder will be equal to 



£("•)'■/(«, /3,r---), 



where a, /?, 7 . . . are what ai, bi, Ci . . . become when we write 

 6h for h, 6 being some proper positive fraction. The demon- 

 stration proceeds pari passu for the generalized form and for 

 Taylor's case of it. Thus, consider IJertrand's proof as given 

 in Williamson's ' Calculus,' second edition, p. 64. 



The lemma upon which the proof depends takes the form, 

 that if/i (supposed continuous between two values of A) has 

 the same value (zero, as it happens in the matter in hand) for 

 two values of A, O/nmst vanish for some intermediate value 

 of h ; which is obviously true, since 8/= i2/8A. The rest of 

 the demonstration remains essentially the same, mutatis mu- 

 tandis, at each point as for Taylor's theorem properly so called. 



The theorem above established easily admits of extension to 

 the case of %, /;,, c^ . . . being the values assumed by a, />, c . . . , 

 when in the quantic (a, 6, c . . . Jj^.t', y, zj^ we substitute 

 OS ^-Jiy-rkz ■\- . . . for x. We may thus obtain a theorem 

 which will bear to Taylor's theorem for any number of vari- 

 ables the same relation as the theorem given in the text to 

 Taylor's theorem for a single variable. 



Since the effect of changing x into x + A + SA may be ob- 

 tained either by first substituting x + A for x and then x -f 8A 

 for X in (f)x, or by a reversal of the order of these two pro- 

 cesses, we obtain the interesting consequence that the two 

 operators 



and 



a-TT +^0-r- + OC -3—3 + 



do do d.d 



dhi dci d,di 



* Consequently, if Qf vanishes, since also (i2 .)f/'will also vanish for all 

 values of i, wo shall have /j =f. It is this ftict of (i2/=0) being the 

 complete solution of (/, =f) which constitutes the importance of the 

 theorem in the Calculus of Invariants. 



