214 PHILOSOPHICAL TRANSACTIONS. [aNNO l/OS. 



bination, and it will be a = m, and a = w, and the whole number of com- 



1- ^. AXA — IXA — 2X&C. . n»xw— Ixm— 2x&c. 



binations= ^ — i. e. = ; — each series 



«X*— 1 X*— '2x &c. nxM— Ixn — 2x &c. " 



continued to n places, and therefore the number of alternations = m X w — i 

 X m — 2 X &c. continued to n places. 



But fully to illustrate this theorem, which, as delivered in general, may seem 

 somewhat too abstracted, to be commonly understood ; I shall therefore subjoin 

 one short example. 



Example. — Let the things exposed be aaabbbcc, or according to our way of 

 notation a^ b^ & : it is required to find the number of their combinations and 

 alternations, taken 4 and 4. 



Then (because in the things exposed, there is no one thing occurs more than 

 thrice, nor more than three things different from each other) will all the 

 forms of combination, which the things exposed are capable of, be these, 



r 3 . 1 J 



viz. ^2.2 > Then 



C 2 . 1 . 1 3 



In the 1st form will jb = 3, 9=1, «=1, |3=1, a = 2, b = 3. 



In the 2d form will /> = 2, , a = 2, , a = 3, . 



In the 3d form will jb = 2, 9=1, a = 1, |3 = 2, a = 3, b = 3. 



The number of combinations in the 1st fortn = - x T = - x - = 4. 



The number of combinations in the 2d form ='*'*"" ^ 



The number of comb, in 3d form = - X . ., , — , « 



« /8x/3 — 1 ix2x 



And the whole number of combinations r= 10. 

 Also the number of alternations in the 



i»*<V.-,« — .. xy " X « - 1 Xn-2x«--3 .v.-* X 3 X 2x1 . ^ . ,« 



1st form = 4 X , fl — = 4 X —^ — = 4 x 4 = 10. 



~ - 3 X 2 X 1 1 



2d form = 3 X = 3 X ,^ = 3X6=18. 



2 X il^ 



3d form = 3 X "- ^-^ Z' V " "^ = 3 x l^iyf^ = 3 X 12 = 36. 



p X p - ir X r 2 X ijl X l2 



And the whole number of alternations = 70. 

 Many are the properties of this theorem in common with others. As, to 

 find the unciae of a multinomial raised to any integer power. To raise an in- 

 finite series to an integer power, though of an interrupted order, without in- 

 troducing any thing immaterial, or which must afterwards be expunged ; and 



