11964} 
Binatibns taken n and n, are produc'd by' the Alternations 
oft the things compofing the parts one amongft the orher : 
A:od therefore the number of thofe = ta the - number of 
thefe = to the number of the Alternations of m things 
taken m and m, the Indices of whofe occurrences are n 
r\A m rt w x i n~i x n*. — 2 x m ->>> - x Sec , continued to m placet 
ana ni— •n ^ x n-i h Scc h ir-n x m-r-i X \c each Seriei cowinucd lon and oimj 
places refpeftively (by Lem, 4th J /. e. becaufe n 4- m— n 
m X ro— I X rr-2. x nt— ; iVr 
n H n- i nn--2 >« n-j dtc 
_ i n xm-^ix rr-ixnz-ajvr. ^^^y^ Serif s cotttittued to n places 
Q. E. D. 
Bat the number of the Alternations in every Combina- 
tion is, =: n >« n — I » n — 2 ^ n — 3 x, &c.. continued to 
n places^ by Lc^. 3d therefore. 
hemma 6th. 
The number of the Alternations of m things abed, &c, 
different each from other, taken n and n, is = m k m — 
jt*. 01 2 H m — 3 continued to n places. Q: E. D. 
Since in the things expos* J the fame things may occnr 
more than once, and alfo n be kfs than m, the Indices of 
^he occurrences which are in fome of the Combinations of 
m things taken n and n, may differ from thofe which arc 
in others ^ but thofe Combinations, the Indices of wh*^ fe 
occurrences are the fame, are^aid to be in^ the ft me form J v 
Therefore- whereas n is equal to the fum of the Indices 
which? are in each Cbmbinatioriitaken n and n, if n be 
txpfeft'd >by all the ditferem Combinacrons of . fuch: Indices 
oniyi (being integer numc^rs) wh'^rcof no one may exceed: 
the> higheft Index of the thirjg& cxpos'd, and being more^: 
tfraa^oneJn^ Combination^ are each of the n; which ai:c^ 
M the. fame Combination, comprehended in a diftinft In- 
dex thereof 5 thefe Expreftiosb cf ii will nectllarily 6e the^ 
l^veral forms'of tte Combinations taken n and n,. where-s 
dEmthingj^ ^ capable r Whence is derived s G^neral^ 
