VOL. L.] PHILOSOPHICAL TRANSACTIONS. 283 



360® 360 360 



of the angles expressed by • X m, 2 X — X m, 3 X — X m, &c. And 



here it may not be amiss to take notice, that the series 1 -\ h 



•^ m ' m + n ' m + 2n ' 



&c. thus determined, is that expressing the fluent of j~; corresponding to 

 one of the two famous Cotesian forms. From which, and the reasoning above 

 laid down, the fluent of the other form, -37^, may be very readily deduced. For, 



since the series ( — — &c.) for this last iluent, is that 



which arises by changing the signs of the alternate terms of the former ; the 

 quantities p, 9, a, &c. will herC; agreeably to a preceding observation, be the 

 roots of the equation z" 4- 1 = O ; and consequently, «, (3, y, (T, &c. the co-sines 



jcQo 180° 180° 



of the arcs , 3 X , 5 X , &c. as appears by the foregoing note. 



So that, making q, q', a'^ &c. equal here to the measures of the angles x 



180" ISO* 



m, 3 X X m,5 X X w, &c. the fluent sought will be expressed in the 



very same manner as in the preceding case ; except that the first term, — log. 

 (1 — X,) arising from the rational rootp = 1, will here have no place. 



After the same manner, with a small increase of trouble, the fluent of 



ji^aiJ ^'r^n "^^y be derived, m and n being any integers whatever. Mr. S. now 

 sets down one example, where the impossible quantities become exponents of 

 the powers in the terms where they are concerned. 



The series here given is i — a:+|--£-f~^— ^1^, &c. = the 

 number whose hyp. log. is — x, and it is required to find the sum of every n'^ 

 term beginning at the first. Here the quantity sought will, according to the 

 general rule, be truly defined by the v^^ part of the sum of all the numbers whose 

 respective logarithms are— px, — qx, — rx, &c. which numbers, if n be taken to 

 denote the number whose hyp. log. = 1, will be truly expressed by TH-f", n-?', 

 N-'^, &c. whence, b y writ ing for/), q, r, &c. their equals 1, oi-\- /«« — 1, a — 

 ^«a— 1, (3 -f /p^^in^ |3 — /(3p— ], &c. and putting d = ^^l—xx,Q'=: 



V 1 — pj3, &c. we shall have - x (n-^* + n-?' + n-'^ &c.) = - into n"* + 



n-'* X (N-^''^~jf n '^~) 4- N-^* X {n-^''^~ -f- N^*^~ + &c.) But 

 jf -^xa/ - 1 -|- N *'^- ' is known to express the double of the ca sine of the arch 



whose measure, to the radius 1, is dx. Therefore we have - into n— *-f n— "^ 



n ' 



X 1 co-s. dx + N"'^'' X 2 co-s. (3'x, &c. for the true sum, or value proposed to 

 be determined. 



The solution of this case in a manner little different, I have given says Mr. S. 

 some time since in another place ; where the principles of the general method, 



00 2 



