4 



GANESH PRASAD, 



In Art. 30 we discuss these two results and attention is drawn to the fact that 

 in Order that approximations be at all possible it is necessary that the peri- 

 phery of the unit area be of restricted size and shape. For example, when the 



area is a rectangle ( -g- X --— j no approximation is possible , X being the length 



of the molecule. Maldng use of (I) and (II) we obtain in Art. 31 the approxi- 

 mate conditions of the phenomenon. These are [A^ , (2?j) and (CJ. {A^ corre- 

 sponds to the principle of energy ; ( . to the condition that as (t) passes 

 from one value to another it assumes all the intermediate values ; and (C'J , to 

 the supposition of the impermeability of the faces. The remaining portion of 

 the exposition is much simpler; in it we find what conditions f{x) must satisfy 

 in Order that the auxiliary function be of Fourier' s type, where f{x) — Y{x, 0) 

 and f{x), /'(.r) and f"{x) are finite and continuous. We show in Art. 32 that, 

 as {A^), and (Cj) are in form the same as ( J.) , {B) and (C), the only con- 

 ditions to be satisfied are those which are necessary in order that (^i), (^J and 

 (Cj) be approximations to the actual conditions. These necessary conditions are 

 given in the end of the article. 



The final result , regarding the condition to be satisfied by /'(.r) , may be 

 thus stated : 



If it be supposed that 



then it is sufficient that 



2%f 



' S ' 



i43ir(:r)| + 676 + f 2 



m 



be negligible. This is the condition (PJ given at the end of Art. 37. Here 



is the quantity which first appears on page 26 and may be called the radius of 



the sphere of influence of any assemblage, A.^ is the greatest length that an 



assemblage can have, /" = — |/ f{x')dx'\, 6,, = — / f"{x')cos 'mx'dx' and b is 



the greatest value of |&„J. In Art. 38 attention is drawn to the fact that, in a 

 professedly inexact theory like the present one, the important thing is the nature 

 of the restriction imposed on f{x)] and, as the quasi -necessary condition (P,) 

 clearly shows , this is purely arithmetical. We conclude the exposition of this 

 theory with Art. 39 in which illustrative examples are given ; in each example 

 we start with definite suppositions as to the magnitudes of A, X^, X^, and find 

 out the corresponding superior limit of error. The table given in the fourth 

 example draws attention to the fact that the error increases very rapidlj^ as 

 the change of initial temperature from assemblage to assemblage increases; in 



