^8 Mr. Herschel on a remarkable Application 



Let B == 0, and {a) gives 



sm. A= 2 . Sin. — . sm. --^— . sm. — -=— . . . sm. * ^-2-—. id) 



n n n n ^ ' 



If A = — , this becomes 



1 = 2 . sin. — . Sin. — . sin. — sm. ■ . (e) 



(6) gives by making B = o, 



COS. A=2 . sm. -i • . sin. --i — - — , . . sin — . ( / ) 



If A = -, this gives 



1 = 2 . sm. ^ . sm. ^ . sm. ^ . sm. -|^ sm. '—^. {g) 



Equation (^) divided by (6). gives, (putting A, for A — B). 



A . TT+A . 27r+A . («_i)w4.A 

 sin. — . sin. . sin. sin. ■ ■ — 



tan. A = . f^] 



sin.|-^-j.sin.^— J...sin.^ ] 



But, to return from this digression, let us take Equation { 8 } > 

 and putting it into this form 



2/».(I+^»)».sin. ^^_,V) 



P (1 — 2^ . COS. i^z + A*) (I 2^ . COS. (w— 4/) -f A^) 



for \|/ substitute each of a series of angles, n in number 



III I 



and let p, p, .... p be the resulting values of p. 



12 n 



The values of sin. [^ — ip) in an inverted order, are (if 

 ,= -{4. + ^.(^)|)sm..;sin.(v + -^);si„.(v + ^). 



.... sin. (v + ^^^ tt), and their product, = sin. — . sin. ^^' 



• • . • sm. 



, , . sin. nl 1.0/^ 



' ;: = F=r- (by Equation {d) ) = -zr-^. 



