VOL. LXVIII.] PHILOSOPHICAL TRANSACTIONS. 365 



n, b the co-efficient of the 3d, &c. and/) the greatest co-efficient: when n is an 



1 /• J \„ abs.wi — aabs. (« — 'i) y. x 4- b ahs. (n — 4) x x mo . » 



even number, (ord. x)n = i Lol — Z -A — zL± — +^ -f i_ . 



V / 2"-' — 2"' 



but when n is an odd number, 



, , X ord./fj — aord.(» — 2) X 1+ i ord. (« — 4) X J; nrpord. x 



(ord.a,)"= ^ -^^ —^^^ , If now X de- 

 note an arch of a circle, by substituting and changing the signs as oft as (ord. xY 



occurs in any of the preceding expressions, we get 

 / ■ X, 1 — COS. 2.1' 



(sin.^)'= ^ ; 



^ . , , 3 sin. J — sin. 3.r 



(sm. xy = — 



4 

 3 — 4 COS. Sj + COS. 4j 

 8 " ' 



(sin. x)* = 



, . ,^ lOsin. a; — 5 sin. 3x + sin. 5a: , ,, . r , 



(sm. xy = -^ ; and universally, if n be 



any number, p the greatest co-efficient of a binomial raised to the power n, a 

 the co-efficient next less than p, b the co-efficient next less than A, and so on : 



1 . 1 / • \ hp — A COS. 2.r + B COS. 4i — &c. , , 



when n is an even number, (sin. x)n = — — — j ; but when 



I , , , . , p sin. X — A COS. 3x + B cos. 5.r — &c. 

 n IS an odd number, (sin. x)n = <—— . 



These series differ from the former only in the signs, and the arrangement or 

 the terms: and when either n, or n — 1, is divisible by 4, the signs remain the 

 same in both. 



16. The reason of the foregoing rule for changing the signs is, that the rect- 

 angle under two ordinates to the hyperbola is always expressed by the difference 

 of two abscissae: and that if from tlie absciss belonging to a greater sector, be 

 subtracted the absciss belonging to a less, the remainder will be affirmative; 

 whereas, if from the cosine of a greater arch be subtracted the cosine of a less, 

 the remainder will be negative. Therefore, that the rectangles, expressed by 

 these remainders, may have the same sign, in both cases, the signs of the 

 remainders must be different. 



It appears then, that the 2d rule, as well as the first, is founded on the prin- 

 ciple of analogy when taken with the necessary limitations, and it is likewise 

 evident from the instances which have been produced, that those rules lead to 

 the very same conclusions which are obtained from the imaginary values of the 

 sine and cosine. There are however instances, in which the analogy between the 

 circular and hyperbolic areas being wholly interrupted, neither the foregoing 

 rules, nor any of the same kind, can be applied; but this occasions no ambiguity, 

 for the construction required in such cases is by its nature restricted to one of 

 the curves only. Of this kind is the Cotesian theorem, which requires the 

 whole circle to be divided into a given number of equal parts, and therefore 

 cannot be extended to the hyperbola where a similar division is impossible. 



