VOL. XXIV.l PHILOSOPHICAL TRANSACTIONS. 213 



a and a, multiplied into the number of the combinations of the ^* taken p and 

 (3 multiplied into the number of the combinations of the r* taken y and y. 



But because p and all lesser indices are comprehended in every index which is 

 greater than themselves ; therefore is a = to the number of />» which are in the 

 things exposed ; and for the same reason, would b = the number of the 9*, 

 and c the number of r' ; but the number of the />*, which are in every form of 

 combination, is = <x ; therefore is b — a = to the number of 9*; also because 

 the number of p* and 9* together, which are in every form of combination, 

 wherein there are 9*, is = a -|- (3 = ^ ; therefore is c — /? = to the number of 

 r*; and so on, how many soever may be the different indices in any form of 

 combination. 



But, by lem. 5th, the number of the combinations of the />*, which are in 



the things exposed, whose number is a, taken a and «, is = ^ Z. \ — TZTo ^^' 

 continued to« places; and the number of the combinations of the y*, whose number 



isB — «,takenj3and|3,is= ° "" * f ° T— Tx /s— 2 ^^* ^^"^^""^^ ^^ P places; 



and the number of the combinations of the r*, whose number is c — Zr, taken y 

 and y, is = ^^^^ — ^ . — &c. continued to y places, a. e. d. 



But every combination, in one and the same form, affords the same number 

 of alternations ; therefore the number of alternations, in any one form, is so 

 many times the number of combinations, as is the number of alternations in any 

 one of these combinations. But, by lem. 4th, the number of alternations in 

 any of those combination?, shall be 



«xn — 1 xw — 2 xw — 3 X n — 4xCT — 5x« — 6x &c. continued to n places 



j> y,p — ixp — 2x &c,|'' ^ q y. q — 1 x &c.|^ x ^ x r — 1 x &c.jy x , each series continued 



i.o pqr, &c. places respectively, a. e. d. 



Now to make an application of this general rule to those particular cases 

 which have already been considered by others, and which are contained in our 

 3d, 4th, 5th and 6th lemmas, and by us more generally demonstrated ; I say 

 if n = w, there can be but one form of combination, and but one com- 

 bination in that form ; and therefore the number of alternations = 



m X m — 1 xwi — Sxm—Sxw* — 4x &c. continued to m places « , 



, ,^ ^ ■^^— 1/3 ^ a, u V «c. each series to p, 



P X p-l X &C.1'* X 9 X 7 - 1 X &c.|^ X r X &c.|y X r* 



9, r, &c. places respectively, i.e. (if/) = l)=mXm — l=w — 2 X m — 3 



X m — 4 X &c. continued to m places, which are the cases of the 4th and 



3d lemmas. 



But if the things exposed be all different, and n be less than m, which is the 



case of the 5th and 6th lemmas ; then also can there be but one form of com- 



