596 MR THOMAS STEPHENS DAVIES ON 



Proceeding thus through 2 6, 3 d, .... n 6 we get at last to 



r"" 1 . . S (a m p m cos ( 1) 6,n) = Sa m . S (q p cos ( 1) w p ) 

 r n ~ l . . S (a m p m sin (n 1) TO ) = 5 m . S (q p sin (w 1) w p ) 



and, lastly; 



r" . . . . S (m cos n B m ~) = S a m . S (cos n Wp) 



r n . . . . S (a m cos n 6 m ~) =Sa m . S (sin n w p ) 



Our business is, in the first place, to find the value of p corresponding to the 

 different forms of n. Now, it is familiarly known that for all the discussions re- 

 garding multiple arcs, all integer values of n may be considered under the forms 

 of 4ju, 4:/j. + l, 4/x + 2, and 4/z+3: but our purpose in the present instance will 

 be effected with equal completeness by considering n to exist under the forms 

 2 v and 2v + l whilst the length of the discussion will be diminished about one- 

 half. 



1. 



The first line has all the even powers of r (r included) : the second has all 

 the odd powers : the third has all the even powers but the lowest, r : the fourth 

 all the odd powers but the lowest, r 1 : the fifth has all the even powers but the 

 two lowest, r and r 2 : the sixth has all the odd powers but the two lowest, r 1 and 

 r 3 : and so on. 



Now, in the case supposed (n=2v) the first line will have j/ + l terms, r be- 

 ing even. But by the general structure of these theorems, the last term is can- 

 celled from both sides of the equation of the porism : whence the number of 

 terms in the first line is v. 



Pursuing the enumeration in the same manner as was done in the preceding 

 proposition, we shall find that : 



the second line has v terms 

 ... third ...... v 



... fourth ...... v 1 



... fifth ...... v-l ... 



... sixth ...... v2 ... 



... seventh ...... v-2 ... 



and so on. 



Again, there will be 2 j/+l lines; for all the multiple cosines are found in 

 them from to 2 " inclusive. Leaving out of view, for the moment, the first line, 

 we shall have 2 v lines, which, in pairs, contain the same number of terms. The 

 last two of these will be 



Lr 2 "- 1 ^ cos (2 v-l) (0-0), 



