290 PROFESSOR WILLIAM THOMSON ON THE 
§ 21 of my former paper, derived from Craustus’s extension of Carnot’s theory, 
we have 
Lid 
where yu denotes Carnor’s function, the same for all substances at the same tem- 
perature. 
Now let the substance expand from any volume V to V’, and, being kept 
constantly at the temperature 7, let it absorb a quantity, H, of heat. Then 
f ee Mi 
Hef Mde=i 4 ff dv. é 7 b). 
V pdt vy? (6) 
But, if W denote the mechanical work which the substance does in expanding, 
we have 
Wt 
Vi pdv . : : : : : €)5 
= (c) 
and therefore 
ira, 
Bsa gg velit bee iotdlh sul, Ht 
This formula, established without any assumption admitting of doubt, expresses 
the relation between the heat developed by the compression of any substance 
whatever, and the mechanical work which is required to effect the compression ; 
as far as it can be determined without hypothesis, by purely theoretical con- 
siderations. 
4, The preceding formula leads to that which I formerly gave for the case of 
fluids subject to the gaseous laws; since for such we have 
; pr=p,v% (1+E4 5 3 Z ' F (1), 
from which we deduce, by (¢), 
Wap, (1+EDlgy . .  . QQ) 
dw Vv’ E 
and Te TE Pot log Cea sy eae . ° . (3); 
and therefore, by (d), 
E 
H= Tas) . WwW . . . . (4), 
which agrees with equation (11) of § 49 of the former paper. 
5. Hence we conclude that the heat evolved by any fluid fulfilling the gaseous 
laws, is proportional to the work spent in compressing it, at any given constant 
temperature; but that the quantity of work required to produce a unit of heat 
* Throughout this paper formule, which involve no hypothesis whatever, are marked with 
italic letters ; formula which involve Boyte’s and Dauron’s laws are marked with Arabic numerals ; 
and formule involving, besides, Mayer’s hypothesis, are marked with Roman numerals. 

