Mr. Jarrett on Algebraic Notation. 93 



Whence, substituting in (l), 



^ CD X 



Ex. 8. To expand i + c.cos x]" in a series ascending by 

 cosines of the multiples of x. 



T+Vcosx\" = 1 + Sjn . — I , by the binomial theorem, 



m I 



oof, ecosa;^'"-' ,, e cos j;^'") ,. ^ ,^, 



CO (• e^™~' 1 1 



= 1 ■\-S,„\[n ., .-— — 2 .S'"r|2OT— 1 ., .cos2r-l.j;' 



2m — 1 I wr—r 1 — 



+ l« •, — • ( „ „ , . jS^I 2 ?/i . I . cos 2rx + \2m. -^- — If, 



L— 2 m V2^"'-' I \m — r 1^ — 2""" .\7n/ > 



2 m I m — r I in Cl_ 



by substitution, 



= 1 + S,,\n .\2m.- 



2in m 



2-"\\2m.\m 



=0 f, e-"'~' „ . , cos2r— l.j; 



2m— 1 t m — r 



,^_. . .,,... . „_ - m-r 



. , e""' „, , cos 2 rxl 



+ ra • ,^ I , . S"'r 2 m . -j [ , 



W2^ .2OT L-— w— r ) 



But 1 2 TO.,- = - — -„; whence^ and by Art. l6. 



g2m+2r— 3 



+ cos2m-l.x.5.[M . |2OT + 2r-3 . ■,„^,,_, 



,, , „ „ , .. ..\2m + 2r-3 .r- I 



2m4-2r— 3 r — 1 I ; i 



