34 MR CLERK MAXWELL ON 



then by Thomson and Tait, § 694, if a is the displacement in the direction of a 



2»<<r + i)-J£ = P-»(Q + B) .... (13). 



Case I. — If R = this becomes 



o ( -\\^ a - d 2 j d 2 G cl 2 G . i\xj) 



^ ' clx dxdy \ dy' 1 dx 2 ) 



Integrating with respect to x we find the following equation for a— 



fc<r + l).= £{^-4g + (.-l)H} + Y • (14), 



where Y is a function of y only. Similarly for the displacement /8 in the 

 direction oiy, 



c* +1 >< 3 =s{!?-<'i£ + (*- 1 > H } + x ■ • < 15 >- 



2n 



where X is a function of x only. Now the shearing stress U depends on the 

 shearing strain and the rigidity, or 



u =»(t+f) < 16 >- 



Multiplying both sides of this equation by 2 (a + 1) and substituting from (11), 

 (14), and (15), 



Hence 



„, 1N d*G d*G . d*G d*G , 1N /^ 2 H d 2 K\ dX dY .,_. 

 2(<r + ^dxW 2 = W~ dxW + dx* + {a ~ VyM + If) + Kx~ + dy~ (l7) ' 



(d 2 d 2 \ 2 „ dX dY „ .( d 2 d 2 \ Tr 



{d^ 2 + dY 2 ) G+ d^ + ^ = ^-^{dx- 2 + df) K - ^> 



an equation which must be fulfilled by G when the body is originally without 

 strain. 



Case II. — In the second case, in which there is no strain in the direction of 

 z, we have 



g = K-<r(P + Q) = (19). 



Substituting for R in (13), and dividing by cr + 1, 



2n^=(l-a)J>-aQ 



d 2 j„ .d 2 G d 2 G „) 

 = dxTyV 1 -^df-°lW +(7K \ ■ • < 2 °)' 



with a similar equation for /3. Proceeding as in the former case, we find 



/d 2 j^Yg dX dY _ a /d 2 ^_\ H 



\dx 2 dy 2 ) dx dy ~~ 1 — a \dx 2 dy 2 ) ^ '' 



