on Converging Seriefes* 
be one value of the given quantity ; the remaining values Can 
be deduced by the fame method. 
In this cafe the given quantity is refolved into a feries 
afcending according to the dimen lions of P, and defcending 
according to the dimenfions of Q; in the former cafe it was 
refolved into a feries afcending according to the dimenfons of 
Q, and defcending according to the dimenfons of P j both the 
feriefes affording the pofibie or impofible parts will always 
converge. 
4. If P — rtr then will (dbPrizP,^/ (— l)) r = P r X 
\iit \iit 
(— I ± s/ ( ~ I = P' X 2 1 ' ( z±= \/ irtv/ — ' h ) r = P r X 2 Ir X 
V 7 (-1); for \ 4 / ( - 1 } = — v 7 i rt\/ - f. 
4. 2. When P = o, or Q. = o, then it becomes the fi rfl cafe 
5. Let P = Cb=i=«, where « has a very fmall ratio to Q ; 
then will (P±:Q / ( — i)) r = (Pd=:P:±:<« y (— i)) r — (P x 
2" X - (~l)) r =P r x 2 zr X s/ ( - 1 )— — X P 
4 r 
l \/ ( - 1 ) X v /( - l) a - 
I — r 
I — r 
> 2 ' x 
I I ’ Tj 
— x x P 
r 2r 
I 
-t- 
r 
r — 2 r 
r X 2 
X 
T — 2 r 
2r X 
X — 1 
X - • — - ■ i_2T xP ' nM^(-i)K 
v v J r 2 r gr 
y ( _ j) 4- &c. In this feries the fame root of the quantity 
& ( — 1) is always to be ufed. 
6. If in the given quantity are contained more quantities of 
the above-mentioned kind or their roots ; then, by repeating 
the fame operation, can be deduced the roots or values of the 
given quantity. 
Vol. LXXVII. L In 
