210 PHILOSOPHICAL TRANSACTIONS. [aNNO l/OS. 



3dly, That m is generally put for the whole number of things exposed, 

 whether all different or not, i. e. equal to the sum of their indices ; and n, for 

 such a number of them, as each combination and alternation must consist of; 

 unless presupposed equal; which explains what is hereafter meant by the com- 

 binations and alternations of m things taken n and n ; or of m things taken m 

 and m; and the like expression, by whatever symbols the number of things out 

 of which the combinations and alternations are to be made, or of which they 

 are to consist, may be designed. 



Lemma 1. — If in a right line, at any distances, be placed any number of 

 things, abed, &c. the number of the intervals ab, be, ed, &c. terminated each 

 by two adjacent things, is one less than the number of things. For, whereas 

 every interval is terminated by two adjacent things, if to any number of things, 

 be added one thing more, one interval only is thereby added, q. e. d. 



Lemma 2. — ^The number of the alternations of m things abed, &c. different 

 from each other, taken m and m, is m times the number of the alternations of 

 m — I things a^c, taken m — 1 and m — 1. For, by lem. 1st, the last letter 

 d, besides the position it has, may have m — 2 positions, viz. in the intervals 

 which are between m —• \ things abc\ but it may also have one more, for it 

 may be put first of all, it may therefore have m positions; and those in all the 

 different orders, whereof m — 1 things are capable ; which being all the possible 

 positions of d, in all the varieties of abc, is all the variety whereof the whole 

 number of things exposed abed, &c. is capable, q. e. d. 



Lemma 3. — The number of the alternations of m things abed, &c. different 

 from each other, taken m and m, is equal to m X m — i Xm— 2XTnX3X 

 m — 4, &c. continued to m places. For let mo, express the number of the 

 alternations of OT things different each from other; m — lo, of wi — 1 things, 

 and the like. It is evident that if m = 1 , it will be ^no = m ; for there can 

 be but one order of one thing. And if m be greater than unity, then will it 

 be, by lem. 2, moz^mXm—lozszmXm— I Xm — 2o=wiX»n — 

 IXwi— 2 X m — 3o =, &c. till we have an equation consisting o( m 

 places; i. e. = m X m — 1 X m— 2Xm— 3X &c. continued to m 

 places, o. s D. 



Lemma 4. — If mu express the number of the alternations of m things 

 a^bPepdieifr &c. taken m and m; and* the number of jb*, p the number of g*, 

 y the number of r«; it will be»»« = 



f)iX»»— lXff» — 2x m — 3xwt— 4XWI--AX kc. conUnued to m places 

 ^xp— ixp — 2x &C.'' X q X 1 — 1 X kc.^ X r X r — i x &c.^ each seriei continued to 

 p, q, r, &c. places respectively. 



For the number of the alternations of any number of things, however divided 



