[ 767 ] 
Cotefian forms. From whence, and the reafoning 
above laid down, the fluent of the other form 
ma y be very readily deduced. For, fince 
the feries 
/ x m 
\m 
x 
+ n 
+ 
x m + 2 n 
x m + z « 
m -f - n 1 m - f 2 n m 3 n 
&c.) for this lafl: fluent, is that which arifes by 
changing the figns of the alternate terms of the 
former ; the quantities p, q, r, &c. will here (agrees 
ably to a preceding obfervation) be the roots of the 
equation z? + 1 = o 5. and, confequently, a , /2, y, S, See- 
the co-flnes of the arcs 2 x c x l8o ° 
(as appears by the foregoing note). So that, making 
'Vji &- c * equal, here, to the meafures of the 
angles — x «r, 3 x — x », 5x^ x &c. tire 
fluent fought will be expreffed in the very fame man- 
ner as in the preced ing cafe ; except that the firfl; 
term, — log. 1 — * (ariflng from the rational root 
p == 1 ) will here have no place. 
After the fame manner, with a fmall increafe of 
trouble, the fluent of ~ 2 Z+x- ma ? be drived, 
m and n being any integers whatever. But I fhall 
now put down one example, wherein the impoflible 
quantities become exponents of the powers, in the 
terms where they are concerned. 
The feries here given is 1 — x 4- - il 41 
x* x* £ 2 2 ’3 
Yfl 2.3.4 ^ C ‘ = number whofe hyp. log. 
