208 THIRD REPORT — 1833. 



using the simple symbol r both in one case and the other. In 

 a similar manner, in the expression of Demoivre's theorem 



(cos fl + V^-i sin 6)" 



= cos {2 r mt -\- n d) -\- ^/ — I sin (2 r n tc + n 6), 



we may suppose n to be any quantity whatsoever *, but r is ne- 

 cessarilj' a whole number. 



In some cases, however, the construction of the formula it- 

 self will sufficiently express the necessary restriction of the 

 values of one or more of its symbols, without the necessity of 

 resorting to any convention connected with their introduction : 



thus, the formula 1 x 2 x 3 r, commencing from 1 , is 



essentially arithmetical, and limited by its form to whole and 

 positive values of r. The same is the case with the fornuda 

 r (r — 1) . . . . 3 . 2 . 1, where some of the successive and strictly 

 arithmetical values of the terms of the series r, r — 1, &c., are 

 put down ; but the formula r {r — 1) ('" — 2) . . . . is subject to 

 no such restriction, in as much as any number of such factors 

 may be formed and multiplied together, whatever be the value 

 of r. In a similar manner, the formula 



m (« — 1) ... (n — r + I), 

 " lT2 77^ r 



which is so extensively used in analysis, is unlimited with re- 

 spect to the symbol n, and essentially limited with respect to 

 the symbol r : it is under such circumstances that it presents 

 itself in the development of (1 + .r)". 



In the differential calculus we readily find 



^ = w (« - 1) . . . (« - r + 1) a:»- % 



and in a similar manner also 



^, (.r" + Ci x^-' + a x^-^ -^ . . C„) = w (« - 1) . . (h - r + l).r»-'- : 

 dx^ ' 



in both these cases the value of n is unlimited, whilst the value 

 of r is essentially a positive whole number ; in other words, 



* The investigation of this formula (like the equivalent series for (1 + x)" 

 when n is a general symbol,) requires the aid of the principle of the perma- 

 nence of equivalent forms, in common with all other theorems connected with 

 the general theory of indices. The formula above given involves also impli- 

 citly any sign of affection which the general value of n may introduce : for 



(cos 6 + V^ sin ^)" = (1)" (cos n 6 + \/'^ sin n 0) 



= (cos 2 rn !r -)- \/ — 1 sin 2 r » ^) (cos « ^ + v' — 1 sin n 6) 



= cos {2 rn "v -\- n 6) + »/ — \ sin (2 r m >r + n tf) 



