1893.] connected with BesseVs Functions. 127 



The first equation is a reproduction of (4) ; whilst the second 

 equation leads to 



2 r°° 



5^0 {'>') = - cos (r cosh x) dx 



TTJo 



r°° 2 r*'^ 



e-rsinh^(^j,_ _ sin (r cos 6') cZ(9 (14). 



■Jo ttJo 







^2 r „,.„„. ., 2 



This equation gives the second form of Y^{r). It will be 

 observed that the second integral on the right-hand side is analo- 

 gous to the integrals by which the first form of the J functions 

 are expressed, whilst the second is analogous to one of the forms of 

 the K functions ; and it is worthy of note that the last class of 

 functions are also capable of being expressed in two different forms 

 which closely resemble those of the J functions. The connection 

 is expressed by the equation 



K^ (r) = £-'■ ''"^^^ ^de= \ cos (r sinh 0) dcf) ; 



Jo .' 



see Hydrodynamics, Vol. il. p. 18. 



4. We shall now consider the functions J^ and F^. 

 By integration by parts we obtain 



P,-, 9.1 ,1 {'^ x^mrxdx 



I (1 — xf cos rxdx = 

 Jo 



2 r 

 whence /, (r) = - 



TT J ( 



rJo {1-xy 

 2 [^ X sin rxdx 



IT Jo (1 -xy^ 

 Now consider 



fze-^'^dz 



j(T+7Yi' 



taken round the rectangle as before. This will be found to lead to 

 the following equations : 

 r" • 1 , ^ci„-hA. 7 1 1 r°° cos(r cosh (i)(Z^ 

 Jo rJo l-|-cosh0 



- ["sin (r cosh </,)r^</,+ j' l^^^II^^Q (15). 



J-.. 2 f^ sill (r cosh. (j))d(fi 2 /"^ i. <\ j> /icx 



^xix) = — — ^^ r^, — ^ cos(?^ cosh rf)) rf6...(16). 



' ^ -^ irr Jq 1+ cosh tt j r/ r v / 



Integrating the first integral on the right-hand side of (16) 

 by parts, and assuming that sin 00 = 0, which is justified by our 

 previous results, we get 



J (?■) = I cosh cos (r cosh (/>) dd), 



'n- J 



