ig Mr, Herschel on a remarkable Application 



I 2^" . COS. (Mw) ^X*" 



If then, n be even, this is equal to unity, as it evidently ought, 

 but if odd 



rO rO) 



J__.j^^^(lr:^r = -(tan.^ . . . . J6-,2J. 

 » » 



2. Let -ar = — , and w = 6w + ^> ^"d we find 

 I ^ 



TiTITT)- (ir;?^) • (t+7=) =tan.-.cotan.V {6,3}- 



I We come nov^ to our 7th Equation, which will afford us 

 results, more complicated indeed, yet equally interesting. By 

 applying the same method of transformation to it, we shall 



find, (supposing cp, cp, (p = (p -f l^^^ — ^, to be written for 



^, and R, R, R to denote the resulting values of R) 



I 2 n 



I, 



an r z 



R R = fl« . - (I +>.->' (!-.-)» ^ 



in i I— 2X" . COS. np -{. a*" (. 2 . 3 1—2^". cos. m (tt + (p) 4- x*" I 



1. If 72 be even, cos. n(p = cos. « (-tt + (p) and, since 1 ^—e* 



I I 



= I — ^ I , this becomes 



R R = a" • _-ilr£>L. . . .{7,21 



I — 2>." . cos. M(2) + X" C ' J 



in , 



2. If « be odd, COS. W(p = — cos. n {tt + (p), whence 



I I 



R R = fl" . 



, "**" „ * 3 1— 2X'' . cos. Wlp + X*" i » . i 1 + 2^" . cos. Nf + X*" ^ 



U 



