580 MR THOMAS STEPHENS DAVIES ON 



already known : but as (except those of the first line, which is clear of the cosines) 

 we shall not require them in these inquiries, it will be unnecessary to discuss 

 them, beyond the extent which our present purpose demands. 



It will be observed, that all the terms, after the first, of the first line, arise 

 from the expansion of the even powers of cos w, and are, in fact, the absolute 

 terms of those expanded even powers. Let these powers be denoted generally by 

 2 fji : then the general form of these terms is 



2^ ' 1.2.3 p. 



This will take, very readily and obviously, the following forms in succession : 



1 1.3.5.7. . ..(2/*-l)x2.4.6 2/z 



2 2 ^ (1.2.3 /u) 2 



1 .3.5.7 (2jn-l)x2*.1.2.3..../A 



V* (1.2.3 fjif 



I 1,3.5.7 



1.2.3 fji 



1.3.5.7 . 



2.4.6.8 2/j. 



Giving to p- the successive values which are applicable to the successive cases 

 viz. 1, 2, .... w, we haye the form of the first line changed to 



For examples of the general expansion, let n=2: then 



fprcos w) 2 = G 2 + r 3 ) 2j9 r cos w + o r2cos 2 &J, 

 2 2 



Let n=3 : then 



(p rcos w) 3 = j0 3 + _ij- 3 (p 2 r + - r*) cos w + - pr 2 cos 2 w - r z cos 3 w. 



9k m 4 



Let w=4: then 



(p r cos w) 4 =jo 4 + 3 p 2 r 2 + r 4 (4 p 3 r + 3 p r 3 ) cos w 



+ (3 p 2 r 2 + - r*} cos 2 a) p r 3 cos 3 01 + 5 r 4 cos 4 w. 



li o 



LEMMA III. 

 To expand (1 cos w)" i multiple cosines ofu, n being an integer. 



This is, in fact, but a particular case of each of the preceding lemmas, and 

 its expansion might be deduced from either of them by making the requisite mo- 

 difications in the formulae. 



