910 



DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



12. Permanent terms for unit element of surf ace traction. 'm,— l. z^z' . 



±=_6i_l(8-,.);3^« + 5i:iIl/3-Tl, . . . 



A,.,4/3-„l/3«. 



^'■• = -4'*'-> 



^"--|^' + 6l''> 



B,..-'^^'-'^V. B,,-L^>+!:^V, '■'..-V'**^ m''''-' ■ 



8 



19: 



COS (o) - 0) 



8 ^ 192 '^ ' ■ 8 ^ 192 '^ ' '■' 8 192 '^ 



Normal traction. G (11), 



- ^ T^^ v^ { (^ - ")(^" - 64) - (11. + 5)(8 - .) I (z - z-)p sin (co - ,o). \ 



The first lines in these values of </>, 0, and \/a lead to 

 ^ = { y (^ - ^T + ^> - '')p' \ *^o-^ (<- - ''0, 



M^ = I -{Z- Z'fp +^r>'^\ COS (o) - O)'). 



The second lines have the forms 



I 2 



a^ Trjx{8 - v) ■ 



which give 



ch, 0,\ , ,, COS , ,J k, B,\ 



j/p = (B + C)(2 - z) cos (o> - oj'), \ 



u^ = - (B + C)(3 - z) sin ((.) - 0)'), > 

 Uz={ - (v+ 1)A4-B}p cos((o - o>'), ' 

 that is to say, 



Mz = { - (v + 1 )A + 2B + C}p cos (o) - (./), 



along with the rigid body rotation 



?/,p = {z- z) cos (w - (I)'), \ 



M„ = - (z -- z) sin (<i> - 0)'), \- . (B + C) 



Wz= - P C0S(<D- O)'). ' 



Here 



B + C = 



1 



1 



_(v2 + 4v - 88) 



= ii iz." ^ v2- 20^ + 92 



a2 I 4 



a2 67r/x(8 - 1/)2' 

 V 2 



and 



)v + 92 ) 



S-v)2 j'i 



7r/u.(8 - v) 37r/x(8 — v) 



-(.+i)A + 2B+c=-i J-^|_r 



(1) 



(2) 



(3) 



(4) 



(5) 



(6) 

 (7) 



(8) 



(9) 

 (10) 



