18 N. F. DUPUIS : 



terms of 6 must have its first term and must coutaiu only odd powers of ^. Therefore 

 assume 



Sin = e-\- a,f)' + «,«"■+ .... a,,. .,(9-"-' + . . . . 



Now take the ndalion 



(Sill H + .sin r^)(8iii " — .sin <p) ~ Sin {ll-\-qt)fim {H — <p) (4). 



Then 



Sin i4 + sin (/j = 0+ tp + ((,{»'■ -}- (p)+ . . . a,„_X(f"-' + qr"-') + .... 

 Sin ^— sin q3=ft—qj+ «,(»'■—(//')+ . . «,,,-,((9 '"' — f//"') + • • • • 



Sin (^4- -■/'.) = ^+'/> + «..('^+'/')'+ • • • «...«-■(^+y^r-'+ • • • • 



Sin {H—rp)= H — qj + a,(/l ~ qjy + . . . a,,,-,^^— '/^)""~' + • • • • 



And wriliug these in (4) and dividing- throughout by ^' — <"/>-, which now becomes a 

 factor, and then putting q) = H the relation 1)ecomes 



I l + <i,'/^+ . . . a,„_,//^"-+ .... } • |l + ofl,«^+ .... (2«-l)a,„^,rt'"-^+ .... I 



= {l + 2-XW-'+ 2---U,,,.,'/-— + } . 



And equating coefficients of 6'"'""" we obtain 



(2-"-- — 2«)a,„. I = 2n \o.,„.^ii..-\-a.„^,a-,+. .... fl„+,a'„_i, «even /' 



^(T„a,„ n Olid 3 



Now by equating coefficients of (l-\ ff^, &c., we readily obtain 



&c. 



Assume that this law of inverse factorials holds up to a-.,,.:-, inclusive. 



Then a.,,,^-^ and «,,„_-, are of opposite signs, and also «j and «, are of opposite signs. 

 Therefore a„,_;iaj and a.,„_:fl:„ &c., have all the same sign for the same value of n. And we 

 readily see that this sign is opposite that of a.^„_i, and is accordingly expressed by ( — )"'\ 



. .^. -nj«,„-i (, ; •(2n)!l(2n — 3)! ^(2n— 5)!5 



I (2n) ! 

 + ^ — L n even: 



+ l^Ml no,ld. • 



" n ! n !i 



] 



C— )"-' i (2-"-2 — 2/i), 



'^ ' ■ (2n— 1) ! ^ '' 



(-)■' 



11 even 

 ^C^", n odd. 



'>y(2) 



a-in-i ■■ 



(2n — 1)!' 



