u^ = 



Elastic Equilibrium of Semi-Infinite Solid. 



Hz 



25 



'^Ttafx ' 



fe-''J,(h-V,{l-a)dk 



+ 



\7tati(XA- a) >' k 



and those of stress are given l;)y 



7r = ^- fl-''llhr)J,{ka)kdk- '^^ fe-"J,(ky)JA[ka)dk 







^ re-'-^Jlkr)J,{ka\lk + ^— P-'-J-{kr)J,ika)^ , 



Tta J ' 7:{X-\- fx)ar •■' ' k 



dd 



Trar J ~(X + ,n)ar •' 



dk 



Ilk 



— fe-''-'J,(kry,(ka)dk, 



zz 



(59) 



= - ^^- fe-"J,ar}J,ika)kdk--^-^ / e-"J,{kr\I,ika)dk, 







7r-= - ^'' fe-'--J,(kr)Jn-a)kdk, 



rd = 0, dz = 0. 



§22. The integrals required liere cannot be evahiated in a 

 very simple way. Some of them are closely connected to the 

 magnetic potential due to a circular current, or to the velocity- 

 potential and stream function of a circular vortex and have been 

 discussed by various authors. In his paper on the inductance of 

 circuhu- coils, '^ Prof. H. Nagaoka has devised a comparatively 

 simple method which may he applied to evaluate all the integrals 

 needed liere. Let us follow his method and descril)e it briefly. 



Pat 



il' = y/ d' — -lar cos 6 + ?•-), (60) 



then we have 



1 /*"' \ 



J^ki-y^ß-a) = -^ / J^,ikB) cosd da, 







-r/7 NT'7 ^ 1 /" " a — rcosä 



J,(ky)J,{ka) = -_-/ J^ikB)- - ^ 



-dd, 



{VA) 



1) Phil. Mag. YI. t; 1903 , p. 19-. 



