•502 The Rev. T. R. Robinson's Experimental Researches on the 



a-al term being 



■ 5 ■ . . ■ 2» - 3 ■ 2w - 1 (26)'-ixr".cos "6i / 2/i + l 2i^ _ \ 

 . 4 . . . . 2n - 4 . 2« - 2 ^ m^"-^ ^ V 2« ^ m' / 



The general term being 



3 



2 



Wheu n is eveu, 



cos" Bde = 2tT I := ^r j X — > , 



Jo il.2. 3... .!«.... 2^ 



and, therefore, the integral of the corresponding term is 



3.5....2«-3.2w-l 1.3.5...n-l 2^ .b"'' .■>•" f2n+l . P 



■n X 



2.4....2w-4.2w-2 1.2.3...irt 



n . W- 



- 1 



and it is derived from that which precedes it by multiplying the latter by the 



factor 



2n-3.2«-l bV 

 X — r : 



n — 2 . n 



(2) 



we thus obtain 



dM= fxF. dc.dl . du . ir 



r b SbrUob' \ 3.5.7.bV/%' , 



m' 2m* V^w' y 2 . 2 . 4 . M« V4m- 

 3.5.7. 9.11. ^»^rV136- 



+ 



-1 + 



2.2.4.4.6.?«" V6m' 

 3 . 5 . 7 . . . 2n- 3 . 2?i- 1 . i"-' . r" /2yi+ 1 . 6- 



-1 



2.2.4....n — 2.M. M'" V nzt' 



This must now be integrated for r. As 7rlr= udu, integrating by parts, 

 [r'du _ -r- nr"-- n.n-2 . r"'* 



5, &c.. 



.It""-' m-1 .m-3 . m""' m-l.m-S.m-d.u" 

 which must be continued till the exponent of r vanishes. Each value of n 



o'ives two sets of terms on account of the factor =^ ~ 1 ; if then we 



o n .71' 



combine the first set of the ?*"* with the second of the (?i + 2)", we have the 

 sum 



PB^^^JJIII^ 



n.n-^.r"-* 



2n+l.u""' 2n+l.2n-l.u^"-' 2n+l.2n-l .2n-B .u 



2n-S ■ 



•&C. 



j^*2 



rr,+ 



w + 2.M="*' 2n+l.M-"*' 2«+1.2w-l.M' 



T^+, 



r, + &C. 



