on Converging Seriefes, 79 
the above-mentioned quantities, that fome may become greater 
than others, and thence may be deduced the proportions above- 
mentioned. 
7. It may not be improper in this place to adjoin a few obferva- 
tions on finding the limits of fome quantities in which others 
contained in given equations become negative or affirmative. 
1. Given an equation involving two unknown quantities a* 
and y ; the limits of the quantity y, between which the quan- 
tity x will become affirmative or negative, may be deduced from 
the following principles. 
The quantity x pafles from affirmative to negative or from 
negative to affirmative, either through nothing or infinite ; or 
from two impoffible roots it pafifes to affirmative or negative 
through two or more equal roots; and, vice verfti , from affir- 
mative or negative to two or more impoffible roots through 
two or more equal roots. 
Find therefore the values of y, when A- becomespo, or 
infinite ; and alfo all the cafes in which two, &c. values of x 
become equal, that is, when its roots become impoffible; and 
from thence can be deduced the limits of the quantity y, be- 
tween which (,v) becomes affirmative or negative. 
2. If = be an affirmative quantity, then P will be affir- 
mative or negative, according as Qjs an affirmative or negative 
quantity, &c. Affiume therefore P = o and Q = o, and from, 
the roots of the refulting equation can be deduced the cafes, 
in which („v) becomes an affirmative quantity. 
3. If more ( n ) unknown quantities (.v, y 9 z 9 v, &c.) be con- 
tained in a given equation ; then, by the preceding method, 
find the limits of (z, v, &c.), between which (*) becomes an 
affirmative or negative quantity, and let the quantities denoting- 
the limits contain net more than («- 1) unknown quantities : 
from 
