Infinite Series . tot 
?L , whofe numerator B is a finite rational and integral fun&ion 
a 7 
of the unknown an Hty : reduce ~ into a feries proceeding 
according to the dimenfions of the unknown quantity • fQ|* 
example, let the feries be A / # r 4B / # r ^ I + C# r "t' 2 * 4* . • . + 
Lx r+Is + + Nx r + l + zs + &c. ; then ( exceptis excipiendis ) 
if s be negative, will the great eft root be nearly; but, if s 
be affirmative, \j ~ will be the leaft root nearly. If / be infi- 
nite, then (exceptis excipiendis , as before-mentioned) the quan- 
tities yj— and \J~~ will be the above-mentioned roots accu- 
rately. 
Thefe principles have been applied to find the remaining roots 
of the given equation as well as the greateft and leaft. 
14. The rule of falfe has been found very ufeful in finding 
approximates to the two unknown quantities contained in two 
given equations, and has been applied to ( [n ) equations having 
(ri) different unknown quantities : for example, it has been ob- 
ferved, that if two or more (in) values of an unknown quan- 
tity (x) are nearly equal to each other and to its given approxi- 
mate value (a/), the unknown quantity v~x -V will afcend 
to two or more (rn) dimenfions in one of the refulting equa- 
tions ; or in more than one equations will be contained fuch 
powers of the quantity ( v) r that if the more equations were 
reduced to one whofe unknown quantity is v r the refulting 
equation will contain ( m ) dimenfions of the quantity v. Hence 
it appears, that in this cafe alfo the convergency of the ap- 
proximate values found will depend on the given approximate 
being much more near to one root than to any other. 
15. When 
