40 GANESHPRASAD, 



(II) that 



H = p{g + %,,E\ |e„|<l, 



the part of J3" due to the temperature /' being assumed to be exactly pg. 

 Therefore it is easily seen that 



JEO \-U, 



Therefore the ratio of 2^(3jEJ+£), the greatest quantity of heat, to be ne- 

 glected, to II as well as the ratio of e, the greatest difference of temperature, 

 to be neglected, to is less than 4i- x lO"'^ or x 10"^ according as k is 

 taken equal to x 10"' or x 10~**, respectively. And , to this degree of 

 approximation, the Solution of the problem is furnished by the auxiliary function 

 V{x, t). In other words, the initial temperatures, Tj(0), T^{0), . .. (0), of the as- 

 semblages being given equal to (ic, J^, {x.^^^'\ . . . {x^^aT, respectively, their subsequent 

 temperatures, T,(t), T,{t),... T^(t) , are given by V(x^^„t), V{x,^„t),... V{x^_^,t), 

 respectively, to the degree of approximation indicated above. 



(ii) Suppose that = 10"^ A„ — < A^ < -^-^ , and A = 3"* x 10"' x Aj. Let 

 Y (X, 0) = fix) = x^ _-|." ^"coy^) ^ 



"where a is a positive constant less than 1, c is an odd integer, and aol + ; 

 f"{x) is Weierstrass's function. Then the left side of (PJ is equal to 



Therefore as well as — is less than 6 X 10~' or 2 x 10 ^ accord- 



ing as A is taken equal to 3~' x 10~' or 3~^ x 10"^ respectively. And, to this 

 degree of approximation, the Solution of the problem is furnished by the aux- 

 iliary function V{x, f). 



(iii) Suppose that A, = 10"^ A„ < ^ZT' ^ = 3"' X 10"' x K\. Let r 



be even ; and, further, let the x - coordinates , a?, . . . x^^^ , of the initial po- 

 sitions of the centres of the assemblages be given by the zeroes of cos (ex) 



where c Stands for the odd number nearest to — , being equal to , say, 



• 75 X 10-^ X A,. Thus 



-^^.H.,o=(^ + i)f 2 = -^,-|- + l,...0,l,2,...|-l. 



