DR MATTHEW STEWART'S GENERAL THEOREMS. 579 



( n -VrS+ .... + t n -i ( r^ n ~ 



-- rj - cos (n- 1) co {<b '_ i r" + ^ * r^J 

 2 r" ," cos w co (<o 4} 



The upper or lower signs of the concluding terms applying as n is even or odd. 



It will be obvious that a slight simplification of this result may be obtained 

 from the consideration that n being integer, io=4, ^=^-1, etc. Our present object 

 will not, however, require any special attention to this circumstance. 



For example, let n=2, then 



(r 2 2 r T-J cos CO + r^f = r* + 4kr 2 r^ + r^ 



- 2 r r 1 cos co [2 r 2 + 2 r^ 2 } + 2 > 3 r* cos 2 co. 

 Let n=3 : then 



(r 2 2 r r l cos co + r l 2 ') 3 =r 6 + 9 r* rf + 9 r 2 / + r^ 



-2 r ^ cos CO {3 r* + 9 r 2 rf + Srf} 



2r* r^ cos 3 CO. 



LEMMA II. 

 To expand (p rcosco)" in multiple cosines, n being a positive integer. 



By the binomial theorem we have 



For the powers of cos co put their values in multiple cosines : then the expan- 

 sion becomes, 



P" + '^ 2 i r + ~~ L ~2 3 + ~ ^J 5 + 



L -^ + ^ + ^ + ^ h .... 1 cos co 



~\ 



I 



cos2w 



f^ 3 /?"- 3 r 3 5^X'~ 5 ^ 21 1 7 p"- 7 ri 1 



L ^- + " Jij |i -- + ~ ^jfr^-* .......... jcosSco 



&c. &c. &c. 



The law of the numerical co-efficients of the several powers of p and r are 



