whence 



TRANSACTIONS OF THE SECTIONS. H 



we thus obtain 



j C03a:»rfr-jl sina^«?a?=cos f 2Ajr+ ^ ) — jsin— ( 2A;jr+ g ); 

 •'o Jo 



and we can equate real and imaginary terms. A curious difficulty, however, here 

 presents itself, viz. to decide what value k (which must be integral) has. In a 

 similar case De Morgan determined the proper value of A; in the following manner : — 

 Put p=co3 ^+/sin 0, then 



Jos 

 e - ■f " <^os « { cos (x" sin 6) ± i sin (.r» smd)}dx 

 



= {cosl(2kn+e)±isml(2kn+d) }rfl+ l"j . 



I e-'"''°'^cos(x"sme)dx=cos^(2k7T+ff)vflMl\ 

 Jo n [^^ nj' 



1 e-^"™'*sin(a;»sin^)rfa;=sini (2k7r+e)T(l+}.\ 

 Jo » V nJ 



Now if ^=0, the last integi-al vanishes; so that we must have k=0, and 

 therefore 



1 cosa;''<?a?=co8^ r( I4. _ 1, I 8ina;»rfa'=sin-^ r| I4- -Y 

 Jo 2« \ nJ c'o 2w \^ nJ 



The above investigation is, however, chiefly valuable as sug/festim of the result ; 

 it contains no indication of the limits between which n must lie that the last written 

 equations may be true and the integrals not infinite. The integrals have also been 



obtained by diflerentiating \ e-ojcsmxdx with regard to a and putting a=0 after- 



wards ; but the results obtained are of the form j a" sin xdx (n integral), and must 



therefore be infinite *. The following investigation of the values of the integrals 

 seems of interest, as it is rigorous and discriminates between the finite and infinite 



values. Integi-ate j J e ' ^ smy dxdy with regard to x first, and we find it 

 integrating with regard to x first, we find it 



=r 



^'^ '^cosec^ (2w>l), 



l+a;2»» 2« 2n 



= 00 (2«<1), 



whence 



j'y-«siny^y.r(l+l) = |^cosecJ 



2w' 



* This method is also given in De Morgan's ' Differential and Integral Calculus,' pp. 630, 

 576. Some analysts (Oettinger, Bidone, &c.) have not seen any objection to 1 x" sin xdx being 

 finite for all values of n ; but unless we are prepared to write with De Morgan (" Theory of 



Probabilities,'' Encyc. Met. p. 436) I eP^dx=— _, because 1 e-P^dx= h, it is difficult to 



Jo P Jo P 



see how this can be admitted. 



