EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 915 

 For a Y force the corresponding expressions can be written down at once. They are 



\mj dy '^ dp/ ' 

 , ( 9 9 3\ _o , 



(16) 



■J 



9x' 'by 9(u', 



We shall work chiefly with component forces in the directions in which /o, w, and z 

 increase, say with P, ^, and Z forces. From (15), (16), (11) we write down : — 



for a P force of \tt\).v units at (p', w', z') 



\ 



3 9 / 9 \ 9 1 



op 



for an 12 force of 47r/xv units at (/>', w', z 



'v 9 92 



^^p' 9co dp'doi 



for a Z force of 47r/xi/ units at (p', to', 2') 



I 1 9 „_, _, 



p 0(1) 



,):..-.-, 



*=a7D.-''-- 



1 '93 r.--' -1 



^ Vp' 9(0 9p' 9(o7^ ) 



, / 9 1 92 \ _, _i 

 ^ 9(o' 9z'^^ ) 



(17) 



15. Auxiliary sources of strain of the types {(p, 0, -v/^) — D'^r'^. 



The results (15), (16), and (17) of the preceding article show that the solutions for 

 a single force subject to given surface conditions may be deduced by diff'erentiation as 

 to the coordinates of the point of application of the force from solutions from sources of 

 strain of the types {(p, 0, \|/) = D^'V'^ 



According to the definition of 14 (9), 



D7V-1 = [ ( - -Vcos k{z - z')GJKn - log 2 + y + log KR}-dK. 



Jo \ nj K^ 



(I) 



Also we have the expansions 



GoiVcR = JfiZxp'OgiKp + 2JjZ/cp'Gji/cp cos (oj — 0/) + .... + 2J„,tKp'G„,2'Kp cos vi{(i> — to') + . . . > 



logKR = logKp - .Zj — ( — ) cos m(to - to'), j ^ ' 



m=\in\p / ) 



where p>p'. 



When these are substituted in the integrand of (1), it is easily proved that the 

 integrations can be performed term by term. Hence 



J)-h' ^ = I ( - -]{cos k{z - z')Z^iKp'G!^iKp - log 2 + y + log Kp}—(h 





-) { cos k{z ~ z')J^iKp'G„^iKp -„-(—) \ -7,dii • cos mUa - to'), | 

 TT/ ( " 2m\p J ) K^ J 



(3) 



where p>p'.* 



The corresponding representation of D^^V"^ in the region p<ip' is of course obtained 

 by interchanging p and p' on the right of (3), 



* The definition of the function G is 



G,ftO = —-. (J _ „iO - e - i»»T Jma). 



2 sm mw 



