50 



GANESH PEASAD, 



phenomenon of the linear conduction of heat in the slab in terms of particular 

 types of ethereal motion. 



Postulating, as in the preceding parts, that the state of things is the same 

 at points equidistant from the central plane of the slab, I propose to show, in 

 the present part, how the phenomenon of the linear conduction of heat may be 

 exactly described in terms of a certain function associated with the points of an 

 everyiühere dense J)ut enumerable aggregate. I proceed, therefore, to formulate a 

 theory of solids which admits of Interpretation as an ether theory, and which 

 may be regarded as thorough-going for the particular end in view : this theory 

 I will call an improperly continuons theory. 



Suppose that with the slab is associated an enumerable aggregate G, of 

 positive numbers, which is everywhere dense in the interval (0, n) and contains 

 jt : I will call G the discriminating aggregate of the slab. Also, suppose that two 

 slabs, and 5^, ordinarily said to be of different materials, say, copper and 

 iron, really diifer only in this , that their discriminating aggregates G^ and G^ 

 are different. [These suppositions may be interpreted as follows: .?j as well 

 as i''^ ether in motion ; but, because of the difference, between the two states 

 of motion, due solely to the difference between G^ and G^, the group of sense- 

 impressions, presented by .Sj and symbolised by and e^, its thermal conductivity 

 and thermal capacity, differs so much from the group of sense-impressions, pre- 

 sented by «2 and similarly symbolised by and c^, that the two slabs are ordi- 

 narily considered as distinct in kind. To use a crude but efPective image, the differ- 

 ence between and s, may be likened to that between 31^ and M^, two mu- 

 sical Instruments, say, open pipes or drums, with different arrangements of nodes 

 but otherwise alike. Now, if and 31^ be sounded in the hearing, but out of 

 sight, of a savage, it is certain that his Impression will lead him to distinguish 

 between iLf^ and in very much the same way as we ordinarily distinguish 

 between Sj and : just as we find it hard to conceive that there is so much 

 common between and as to make an interchange of thermal conductivities 

 and thermal capacities between them possible, so the savage will find it hard 

 to conceive how Jij and 31,, may be made to interchange their musical properties.] 



Let I be a variable with the aggregates — G and G as its domain ; also, as 

 in the first part , let T(x, t) stand for the temperature at tinie t at any 

 point in the slab whose - coordinate is x. Now, let C(|, t) be a function of | 

 and ^, even in | and connected with T{x, t) by the equations 



T{x,t) = ^/ ' C{l,t)dl 0<x<7t-[, 

 T{x, t) = \rrvf ^ 0 ^1, %-\<x<%, 



l being a constant dependent on G. I will call C(^, t) the char acter istic funcdon 

 of the slab : this is the function alluded to above. Also, taking a centimetre as 

 the Unit of length, I will suppose, for the sake of fixity of ideas, that l < 10"'. 



