ALGEBRAIC FORMULA. 25^ 



equation by 2<p cof <p t and take the general equation of the 

 fluents, we get 



/^cof<p(V-H> 2 — 2abcof<pY=B<p^2A+C){m<pi- , -(B-\-D)Cm2<pl-&:c.-, 

 therefore, when <p zz sr, 



•s-B zz /2<p cof <p(a 2 -f b 2 — lab cof <p)n. 

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

 2<pcof 2<p, and take the relation of the fluents as before, we get - 

 ^C = /2(pco£2(p(a 2 -j-b t — 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 afTu- 

 med equation 



(a* 4. £ * _ 2fl£ cof <p) n zz A -f- B cof <p -f C cof 2$ -f- &c. 



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

 vity, let us put A for a ab ; thus we get 



— 2n{in? B fin <p -f- 2C fin rtp + 3 D fin 3^ -f &c. 



A — 2cofp A + B cofp + Ccof 2p + D cof 30 + &c. 



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

 tiplied by the denominator of the other f>de, fubftitutmg 

 fiti(p -f i)p-ffih (p — i)<pfor 2 cof <p fin /xp, and, fin (/> + i)<p 

 — fin(/> — i)<p for 2fin<pcof/<p, then, putting the refult zz o, 

 we get 



K k 2 +AB 



* Traite du Calcul DifFerentiel et du Calcul Integral, par Lacroix ; vol. ii, 

 page 120. 



