071(1 Cosines of Multiples of a variable angle. 261 



cos fl + cos(fl + /0 + cos (5 + 2A) ■!-.... + cos (fl + («--l)/0 



nh 

 sin — 



cos(fl4.<'M_n- \ — 



sin- 



{^Hn-l)\) 



sind+ sin(H^) + sin(fl + 2^)H-.... + sin(d+(«— 1)^) 



> (a-) 



= sin (, + (,_!)_)_- 



sin- 

 2 



are formulae which will be needed ; and we will begin with 

 (p(a?) = Bo + BiCos;r+B2Cos2a?+B3Cos3a;+ . . (A.) 



(p(a) = Bo + Bj cos « + Bg cos 2a + Bg cos 3« + ...... 



f (a + ^) = Bo + BiCOs(a + ^) + B2COs2(a + Jc) + B3COs3(a + k) + ... 



<p{a + 2^) = Bo+ Bi cos (« -f 2>?r) + Bg cos 2(a + 2k) + Bg cos 3 

 (« + 2/<:) + ... 



^(«+(n— l)^") = Bo + BiCos(a+(r«-l)^) + B2Cos2 

 (« + («- 1)^) + B3 cos 3 (« + («- J)^)+... 



Adding these together by summing the terms vertically by 

 means of the first of (a.), putting ix for 5 and ik for h, we have 



. nik 



i;^(« + i'^)=wBo+5;B,cos?(«+(«-l)|) ^- • (!•) 



sin— • 

 2 



Let B;„ be the last coefficient which is sensible, then i will 

 represent all the numbers 1, 2, 3, .... m. In order that all 



the quantities B, may vanish, we must have— = w, or ^= — . 



Then . nik . . 



sm-rr- = sm«7r=0; 

 2 



but 



. ik . iv 



sin — = sm — 



2 n 



is not nothing if « > w, or greater than the greatest value of /. 

 If/t=2«, 



cos? 



(, ^. k\ . . ink . , 



«+ (w — I) — 1= cos 2«a, sm — - = sm tnu; 



