ni~2 X m— 3 O &c. till we have an Equation eon- 
fifting of m places 5 i= m x m — i k m— 2 x m-—^ ^ 
eontinu'd to m places. Q/, E D. 
Lemma 4th. 
If ni » Exprefs the number of the Alternations of m 
things ap b cp e^f dv. taken m and m, and the noaiber 
of P'? i3 the number of q% 7 the number of r*, jt will be 
«,-, — i^iym— T X m— 2 X m— ? x,_ ra— 4 x iR— ? x g^c, coniiou d to m nbca. 
ill — ™ ZZ:^-^ Q . n rrz: 
P p-i X p-2 X &c.«^ K q X q.. 1 X Scg. ^.^ r r-i x 3cc, each Serietcontiauetl £» 
Pt>ci.T, d^c. places refpedively, 
- For the nna}bcr of the Alternations of any number of 
things, however divided into parts, is produced by a con- 
tinoal Multiplication of the Alternations of thofe things a> 
mongftthemfelvesrefpeaively.which compofeeach part,into 
th,e number of their Alternations oneamongft the other^ /. e. 
in theprefent cafe ( thefeveral occurrences being fuppofed 
to qompofe thefeveral parts, and confequently the number 
of the Alternations of the thingscompofing each part equal 
to unityj m « = to the number of the Alternations of the 
things compofing the parts one amongft the other ^ but the 
number of their Alternations one amongft the other, is the 
fame in this cafe, as if the things exposed, being all different, 
were divided into the lame parts ^ for the things which 
compofe each part in bo{;h cafcs^ gre diffi trent from the reli 
of the things expos'd • ^ ' - " t/qd. 
m X rr— T X ni x p-" - ^ ^ r - x ,-''t. ' T. ri'ntTt^ d tC m placCl 
m • . j/^- 5" , , J> , ... _^ 
p X p- 1 X p.. 2 -^.^ q-* I x '&,c. |r x. r x r-- 1 x Scc K eac?? Scrfet continued 
to p, q, , r places re(pec5rvely.- * Q. 'I:!!. u\ 
Lemma.' '^th: 
-^The number.of the Combinations of ;ra things a bed, 
i^c. different each from' other, laken n and. n, is equal to 
m..ixro.,2xrn-.jx &c. ^^^j^ Serics coutiriued to n fil^iceV 
Tor if the things expGs4 be divided; io ^twp./^parrj, viz. 
tri che' r4twof n and mT^o,^ 'tis evident il^t tueir difP^'F^t 
A a aaaa aaa-aaa 2 Com- 
