( ^0/^7 ) 
But every Conibinutiou^ in one and the fame, form, af- 
fords, the fame iiumbri of Alternations : Therefore the 
number of Alrernalions, in any one form, is fo many times 
tlie number of Combimtions, as h the number of Alter- 
tiatxottS'in any one of thefe Combinations, < 
But (by le^A 4th) the number oi Alternations in any 
bf thofe Combiijenons (hall be 
n X n-i X n-2 x n^-^ x r-4 + n-5 x n-g x S<c, continued ton place» 
p X 1 __i X X Scc l*^ >« qxq-2X&c.|^ ^ » X r~i X &c"l^ '* each Series co:^cinued to p qr &c, 
|)laces refpedively. Q. E. D. 
Now to make, an application of this general Rule to thoft 
particular cafes which have already been 'confider'd by o- 
thers, and which are coutain'd in our 5d, 4th, 5th and 6th 
£e;5^/^.zV, and by us more generally demonftrated 5 I fay 
if n ~ m? there can be but one form of Combination, 
and but one Combination in thar form 5 and therefore the 
■ m X rn—T x m-2 x m— j xm.r-4. ^ ^-.m-ir'n tr> m r;t<cr $ 
number of Alternations .= ^r^rr^..^^ «^Tjm^,! ^ rT&T^- 
d^^*. eacli Series r(5:p q r, dv. places refpeftively, i. 
(if p— .i)=:m X m— I « m—a x m— 3, x m-4 ^ d^c- 
contmucl to m plasms, which are theVales of the 4th arid 
3d Lemma s* rr ^ 
^ % But if the things expos d are ail difterentv and nbe lets 
thanm, which is. the cafe of the 5th and 6rh Umm.xs, 
then alfo can there be but one form" of Combination, and 
\Xt will be A m & — n, and the whole number of Com« 
binations '==-^~r^TZ^I7^'c\ '"n x n— 1 - x^^'^i-iy c 
each Series continued to n place?, and tberefore (he nufc^ 
:ber of Alternations m x m— i x m-~2 « &c. continu d 
to n places. . 
v Bur fully to liluftrate this Theorem, which, as deliver- 
ed in o-eneral, may feem fomevvhat too Abftrafted, to be 
commonly underftood; I (hall fubjoyn one filiori Exam- 
ple. 
