VOL. LXXXVIIl/J PHILOSOPHICAL TRANSACTIONS. 413 



sin. z -f- bz + 4- b . sin. 2z + c (sin. z -f- -J- sin. 3z) + d (i sin. 2z -f- ^ sin. 4z), 

 &c. ; which equation, when z = 3-J415Q, &c. = tt, becomes/—— ,^- = barely 

 bz = Btt, the sines of z, 2z, 3z, &c. being then = O. 



15. Now it appears, by the notation in art. 4, that _ = (a + b)~ n X 



^r — ; \, and that x = ^ ± — — ; we therefore have, by proper 



substitution, 



2xz — 2xx ■ 2a 2-lv 1 



X 



(a — bx) n •" V(l - ») (« — **)" *"(« + *)" V(l - w) ( V(w — cc) 



- 2 2^ 3 - 3 " 



6 (a + 6)*'- 1 X V(l — vv) */(yv~— *) 



of which 2 fluxions the fluents may be found, when n has any particular value. 

 16. First, let nbe|; then the last expression in the preceding article becomes 



X ... ffl. _, - r-^rr X „,_ " _, v Now, the fluent 



b(a + 6)| V(l — ct) V(w — cc ) b ( a + *)I -/(l — w ) Vw — «w)' 



2o 



of the affirmative part of this expression is evidently = — X the fluent of the 

 fluxion in art. 5, that is, = — A?r , and the negative part, by converting -/ (l — vv) 



— — 2 2*w VV Sv^ 3 *>i?^ 



into series, will become ^- + -^ X £„_«) + T + 2 . 4 + ifi^ *c.) ; 

 the fluent of which appears, by art. 5, to be f^—rj)! ( a 4" £ + ~r + rV + 

 I* . u' * 3 &c), which will vanish when v = c, and therefore needs no correction ; 



4.0.8 



and after further transformations and reductions, for the value of the co-efficient b, 

 the ingenious author obtains the following expression : 



r 32 - lOcc 



B = ^ A £__ x )l6-9cc " 



6 «H« + *U ^ + ^(x _ cc ) ( p _|_ ^ _f_ TC 4) ? which is its yalue when 



n = ■§-. 



17. We are next to find the value of this co-efficient, when n = -§-; which, for 

 the sake of distinction, he denotes by b'. With this value of n, the fluxionary ex- 

 pression in art. 15, becomes 



2a_ 2-:t>~ 4 2_. 2w~' ... , . 



6(0 + 6)« X -v/(l - to) -/(to - cc) 6(0 +~6)f X -v/(l - vv) */{vv - cc) 5 WniC " Dem S 



compared with the fluxions in art. 5 and 10, it will appear that the fluent of 

 the former part, when v = 1, is = — aV, and that the fluent of the latter part is 



== ' T~ A?r ; which fluents, taken together, are, by art. 14, = bV. Therefore we 

 have b = -r a — j- a = - (a a — - a). 



3dly, to find the Values of c, d, e, &c. 



18. The values of the co-efficients a and b being n6w found, corresponding to 

 the values of n, -§- and 4, we might proceed in the same manner to find the value 

 of c. For, if the equation in art. 2, be multiplied by 2 cos. 2z, and cos. (m — 2) 

 z +• cos. (m + 2) z be written for 2 cos. 2« X cos. ws, it will become 



