ALGEBRJIC FORMULA. 25^ 



equation by 2<p cof f, and take the general equation of the 

 fluents, we get 



y2^cof(p(a^4-^'— 2rt^cof?))'=B^-j-(2A4-C)fin^-f-j(B+D)fin2?)-f&c.; 

 therefore, when ?> = sr, 



Again, if we multiply both fides of the fame equation by 

 2^cof 2ip, and take the relation of the fluents as before, we get - 

 ?rC=:/2(?)Cof2(p(^^ +^' — 2abco£(p)\ 



4. Proceeding in this way, we might get a fluxionary ex- 

 preflion for each of the remaining coefficients D, E, &c, but 

 this is not neceffary, for they may be all found from the firft 

 two, A and B, and from one another, as we have already obfer- 

 ved ; and the fcale of their relation has been determined as 

 follows * : 



Let the fluxions of the logarithms of each fide of the aflli- 

 med equation 



(rt' + Zi» — 2ali cof<?)" =: A +B cof (p + Ccof 2^ + &c. 

 be taken, and the whole be divided by <p; alfo, for the fake of bre- 

 vity, let us put A for ^—^ — ; thus we get 



— 2»finp B fin ip + 2C fin 2tp + -^D fi n 7,(p + &c. 



A — zcoi(p~ A + B cof p + C cof ip + D cof 3? + &c. 



Let the numerator of each fide of this equation be now mul- 

 tiplied by the denominator of the other f^de, fubftituting 

 fin (/' + i)?'+ fill (/>— i)<pfor 2cof<pfinp(p, and, fin(/)+ i)p 

 — fin ( /» — i)ip for 2 fiu ^ cof/(p, then, putting the refult = o, 

 we get 



K. k 2 + A B 



• Traite du Cakul Differentiel et du Calcul Integral, par Lacroix j vol. ii. 

 page 120. 



