1 26 Professor Booth on a new TJieorem in the 



different, so Poinsot, on the other, having shown that the 

 methods pursued were fallacious, and the results themselves 

 false, has given the true development by a method complex 

 in the extreme, founded on the. principles of a higher analysis 

 than is generally introduced into such investigations. 



It appears somewhat strange, that even since the correct 

 series has been given, the erroneous developments should 

 have been retained in some recent publications on the subject. 



In the following pages the method pursued will be found 

 different from those generally adopted in the investigation, 

 no principles being assumed beyond such as may be deduced 

 from elementary algebra; while the method of determining 

 the coefficients of the expansion, by the aid of the fundamental 

 theorem first established, will, it is hoped, be considered, 

 simple. 



Let a, b t c, d . . . I be any m numbers, positive or negative, 

 integral or fractional, or even imaginary, m being less than the 

 integer n, and h being any finite number whatsoever, then we 

 shall have the following theorem : — 



= {a,b,c i d...l} — n {{a — h) (b—h) (c — h) ...} -, 



11. n— 1 ,, o7 \,, qm, a 7N ., nn — l.n — 2 

 + 12 {(«~ 2 h ) (*- 2 k ) ( c ~ 2 *)♦•*}- f23 M.) 



{(a-37i) {b-3h) (c-3h) } &c. &c. J 



or, separating the symbols of operation from those of quantity, 

 we may write the theorem thus : — 



0= (1 - if T [(a - ih) (b - ih) (c - i k)...(l - ih)] y 



the symbol I" denoting the sum of all the values which the 

 quantity within the brackets takes, when i is made to assume 

 every integral value from to n inclusive. 



This theorem may be proved as follows : Let the sum of 

 the series (I.) be = V, assume the arbitrary quantity k a mul- 

 tiple of h, and let a, /3, y, S . . . X be the differences of the m num- 

 bers a, b, c, d...l and/r, so that a = k— a, b = /c—(3 t c^fc—y, 

 d = k — 8, . . . / = h — *> then we shall have the equivalent ex- 

 pressions 



{a,b,c,d...l} = k m + Ak m - l + Bk m - 2 MA + N. 



In the same way, if k' = k — h, we shall have 



(a-h) ib-h) {c-h)... = k' m + AJr- l + Bk' m - 2 ..MV + K. 

 Again, let k" = h — 2 h, and we find 



(a -2A) {b - 2/0 (c - 270 m k" m + AF" 1 ' 1 



+ Bk" m - 2 MF + N. 



