of equal 07'der and argument 



43 





Jnin)=^ 





7r2*3-^ 



2. In order not to restrict ourselves to the case in which n 

 is a positive integer, we take the Bessel-Schlafli integral*, namely 



sin IITT 



J II (^) = - cos {nd — X sin 6) dO — 



,-nd -X sinh ^ 



dO, 



(which is valid whether n be an integer or not), and, after writing 

 n for x, we integrate by parts. This process gives 



Jn (it) = ^ 



d 



nir j 1 "" cos 



sin mr 



-^ -Tj. {sin n (0 — sin 6)] dd 



+ 



d 



IT J 1 + cosh 9 dd 



|g-«(^ + siuh^), ,^ 



nir 



sinw(^ — sin^) 



1 — cos 6 

 1 



+ 



sm nir 



-n{0-^sm}\d)-\ ^ 



1 + cosh^ 



+ 



'^smn(^-sm^)^^^^^^^ 



/, 



mr] Q (1 — cos 6)" 



sinnTT p sinh^ ^-n(d + s.\nhe) ^n 



"^ TT Jo (T+COsh^)^^ ;''^' 



The integrated parts cancel ; and 



^^^^^ -n[6 + ^mh0)^0 ^ f"(l+C0sh6^)«-»(^ + «i"l^^)(/^ 

 (1 + cosh^)- Jo 



= lln; 



and so, when ?« is large and real, 



r y . 1 f '' sin ^ sin w<f> , , „ , ^, 



?i7r.lo (1 —cose')* ^ 



where </> has been written in place of 6 — sin 6. It is obvious that 

 <l> inci-eases steadily from to tt as ^ increases from to tt. 



When 6 is small, (f)r^^6^ and sin ^ . (1 — cos 0)~^ ^ 80-'^. Hence, 

 as ^ -> 0, • 



<^^ sin 8 



(1 — cos 0y Qi ' 

 Now write | (60)* sin ^ . (1 - cos 0)~' =/ (</>) ; 



Schlatii, 3Iath. Ann. m. (1871), p. 14«. 



