VOL. XXIV.] PHILOSOPHICAL TBANSACTIONS. 21 I 



into parts, is produced by a continual multiplication of the alternations of those 

 things among themselves respectively, which compose each part, into the num- 

 ber of their alternations one among the other; i. e. in the present case (the 

 several occurrences being supposed to compose the several parts, and con- 

 sequently the number of the alternations of the things composing each part 

 equal to unity) mu = to the number of the alternations of the things com- 

 posing the parts one among the other ; but the number of their alternations 

 one among the other, is the same in this case, as if the things exposed, being 

 all different, were divided into the same parts; for the things which compose 

 each part in both cases, are different from the rest of the things exposed; i. e. 

 by lem. 3d, 



fflXOT— lx»i— 2x>»— 3xm — 4xffl — 5x &c. continued to m places 



pxp— Ixp— 2x &C.1* X 9 X 9 — 1 X &c.h Xrxr — Ix&c |> each series con- 

 tinued to p, q, r places respectively, a. e. d. 



Lemma 5. — The number of the combinations of m things abcd^ &c. different 

 from each other, taken n and n; is equal to »^ x m - i x m - 2 x m-s x &c 



' ^ « X « — 1 X n — 2 X » — 3 X &c. 



each series continued to n places. For if the things exposed be divided into two 

 parts, viz. in the ratio of n and m — n ; it is evident that their different com- 

 binations taken n and n, are produced by the alternations of the things com- 

 posing the parts one among the other: and therefore the number of those = to 

 the number of these = to the number of the alternations of m things taken m 

 and m, the indices of whose occurrences are n and m — n = 



mxm — Ixm — 2x»i — 3x &c. continued to m places 



n x. n — 1 X &c. X rn — n X m — «— 1 x &c. each series continued to n and m — n places 



respectively by lem. 4th, i. e. because n -{■ m — n =. m, = 



mxm— 1 X m — 2x»ra— 3 &c. , . i.- j *^ 1 



— r-5— each series contmued to n places, q. e. d. 



n X n— \ X n — 2x«— 3 &c. ^ 



But the number of the alternations in every combination, is = n X n — 1 

 X n — 2X n — 3 X, &c. continued to n places, by lem. 3; therefore, 



Lemma 6. — ^The number of the alternations of m things abed, &c. different 

 from each other, taken n and n, is = 7»Xfw — \Xfn — IXm — SX, 

 &c. continued to n places, a. e. d. 



Scholium. — Since, in the things exposed, the same things may occur more 

 than once, and also n be less than m, the indices of the occurrences which are 

 in some of the combinations of m things, taken n and n, may differ from those 

 which are in others ; but those combinations, the indices of whose occurrences 

 are the same, are said to be in the same form : therefore, whereas n is equal to 

 the sum of the indices which are in each combination taken n and, n if n be 



E £ 2 



