Dr. Wahing on 
x- nor u-y, nor $ - y, &c. are whole numbers : let m be 
the greateft of the indices m, m', m" , &c. ; n the greateft of 
the indices n, n', n", &c. ; r the greateft of the indices r, r', r'\ 
&c. ; then, from the fums of the feriefes given, whofe general 
terms are 
(*+*)“ (»+ ’ (z.+ ’ 
(?+“) 
(z+p)” (z+i ?) 1 
+ |3 (z + y) r (z + y ) 1 
:? • • 
? &c, cau 
be deduced the fum of the above feries, whofe general term is 
given above ; multiply each of thefe terms into unknown co- 
efficients e',J g, h, & c. ; then reduce them to a common deno- 
minator, whichds the fame as the denominator of the given , 
general term, and add them together, and make the correfpon- 
dent terms of the fum refulting equal to the correfpondent 
terms of the numerator az! + bz 1 — 1 4-icz 7 — 2 4. &c. of the given 
general term ; and from the equations refulting can be deduced 
the co-efficients /,/, g } h, &c, aad thence from the given fums 
the fum of the feries required; 
Approximations to the fums of the feriefes may be deduced ; 
from the methods given in the Meditatlones Analytic*#, The fum 
of fome few cafes have been given from the periphery of, the 
circle: for example, when a and ware whole numbers, and; 
£=1; or, more particularly, when m~ 2 and e = i, and 
fome other particular, cafes,, which may. be with nearly the 
fame facility calculated from approximations; the cafes given 
indeed are fo few-, unlefo when e = 1, that they can very rarely 
be applied f 
1 7. If the dimenfious of % in the numerator be equal or'greater 
than its dimenfions in the denominator ; that is, / be equal or 
greater than m +.m / +,m // + See. + n +V +n' / + &c. +r+r f 
~br- 
