62 GANESHPRASAD, 



it is easily seen that ca' (x, t^) is a continuous function of x , whatever v 

 may be. 



sr _ 



Now, let J{t) stand for f {x' , t) dx' . Then, representing by a 



positive quantity such tbat t^ + t^ belongs to (/, 



J{t,+ t,)-J{Q = r \ai\x',t +l)~(o\x',tM dx'. (3) 



Remembering that co' (x', U + ty) + &' (a;', t,) is continuous in a;' and, conse- 

 quently, integrable, it f ollows that, if q {x, ty, t, ) Stands for 



/ \Gi{x\iy-\- t^ — (a[x\t^\dx\ 



— TT * ' 



J {ty J(ty) [ I OJ {X', ty + iy) ^ (O {x\ t,) | ^ {x', ty, ly)] ^ ^ 



\a)'(x',f^ + Q + co'ix',t,)]q(x',t,,Qdx'. (4) 



7t 



But it foUows from the impermeability of the faces of the slab that the 

 total quantity of heat in it remains unaltered. Therefore 



q{jt,t^, g = 0; 



and, consequently, (4) becomes 

 J (ty + g - J{t,) = - \m' (.i-', ty + g + oj' {x\ t,) I q ix', t,., ^7) dx'. (5) 



7t 



Now take t^ < t^. Then, whatever a;' may be , 



ca' ix',t^ + T,) = C3'{x',t,)+2fi6, 



q(x\t„t;) =f^^^'^co'{x',t')dt' = ^;ja9'(.x'',g + 2ö,(j|, | e|, | ej<i. 



Therefore (5) becomes 



J {ty + K) - J{t,) = -2Ty f {x', t,) Ydx' + 4< 6,ö (3ö, + 26) , (6) 



— 7t 



where (Oy Stands for the greatest value that | cj' {x', ty) | can have and | | <1. 

 Now, since coy is finite , 43t(? (3o3,, + 26) can be made as small as we please by 

 choosing 6 and, consequently, t,. sufficiently small. Therefore it foUows from 

 (6) that 



