!<! Mr, Herschel on a remarkable Application 



the axis and » odd, or, one inclined at an angle -^, and n of 



tlie form 4W + 2, or, lastly, where 5 = -^ and ;i =: 6w -{- 3. 



I 



4. Let COS. n^ = — , then 



JO /') '« O+x')" _ n (i-fx*)-.(i4x') f •) 



IK i ^ J 



(7) 

 This takes place whenever n is of the form {6m-\-i) , — y-, 



as when 5 = ~ and n =z (6m + 1); Q =z ^, and n = i2w 

 + s, &c. ^ 



5. Let cos. nQ = ^, then 



^ ^ — r • iJ-x"+A»« — -r • i^^tn • • li>i*C. 



Here n must be of the form ^ . (3^ + i)> and if 5 =:= — ^, 



« is of the form ^m + 1 • 



We will now proceed to our second equation {si, and by 

 an operation exactly similar to that from which we obtained 

 the equation 1 1,1 1 , we shall find 



I — 2tf" . COS. nO+«*" 



r r^' — ai^i-^K*) . ,_,,.. COS. ^9+^- ' • • • {2>U 



I K 1 *■ ■* 



I— 2e" . COS. «9 + tf'" a"! I— 2?" . COS. n9+e'» 1 



a \^A-^-A j • ;,n^;,— » _2 . COS. «9 ~~ n(e + «} 71(9-*)* 



4 . sin. — . sin. 



2 2 



This transformation in a, however, as it possesses no particu- 

 lar elegance in point of form, and much complexity, we shall 

 henceforward omit, except in a few remarkable instances. 



