20 Mr, Herschel on a remarkable Application 



-x^)"(i- 



^ K = a . r— rr. {7,10}? 



I n 



but, if « = 6 m + ij 



^ R = ''" • ^^rri^) • • {7."}- 



These are always imaginary expressions when e 7 1 and n 

 odd. In fact, R, in the hyperbola, must be written 



instead of 



v^e* . COS. <p'^ — I V I — e* . cos. 



now v/e* — 1 = ^ . j^^- Thus, this expression, like the rest, 

 is easily transformed, in functions of ^, x, x', the x' is now real, 

 and the part involving x will always be of the form/(x"*4:X~~^), 

 and therefore readily expressed in trigonometrical functions. 

 Before we proceed farther, it will be necessary to premise 

 a transformation of Cotes's formula, which we shall have oc- 

 casion to make use of. It is as follows : 



sin. (A + B) . sin. ( A — B) =: 2^"""^ • P • Q, where 

 P = sin. [^] . sin. \Z±L^-^] . sin. (!I±^±1>) .... 

 sin. (("-"')-+(A+B)| 



Q = sin. (-J-) . Sin. (-^7—) . sm. [-^^ ) ... . . 



(«-l)7r + (A — B)l 



>{a) 



sm. 



The demonstration is extremely simple, 



1 — 2X . cos. a -\' X :=(i-— ex. cos. — •\' X ) (1 — 2X . 



cos. ^ i- X ) (.1 — 2X . cos. ^i^ ^ + -^ ), 



or dividing by at" 



x" + x~^ — 2 . cos. a= {x + x"^ — 2 . COS. -~J 



