272 PROFESSOR WILLIAM THOMSON ON THE 
my former paper, constitutes the second part of the paper at present com- 
municated. 
Part II.—On tue Motive Power or Heat THROUGH FINITE RANGES 
OF TEMPERATURE. 
24. It is required to determine the quantity of work which a perfect engine, 
supplied from a source at any temperature, S, and parting with its waste heat to 
a refrigerator at any lower temperature, T, will produce from a given quantity, H, 
of heat drawn from the source. 
25. We may suppose the engine to consist of an infinite number of perfect 
engines, each working within an infinitely small range of temperature, and 
arranged in a series of which the source of the first is the given source, the 
refrigerator of the last the given refrigerator, and the refrigerator of each inter- 
mediate engine is the source of that which follows it in the series. Each of these 
engines will, in any time, emit just as much less heat to its refrigerator than is 
supplied to it from its source, as is the equivalent of the mechanical work which 
it produces. Hence, if ¢ and ¢ + dé denote respectively the temperatures of the 
refrigerator and source of one of the intermediate engines; and if g denote the 
quantity of heat which this engine discharges into its refrigerator in any time, 
and g + dq the quantity which it draws from its source in the same time, the 
quantity of work which it produces in that time will be J d ¢ according to Prop. 
I., and it will also be g » d¢ according to the expression of Prop. II., investigated 
in § 21; and therefore we must have 
Jdqgq=qpdt. 
Hence, supposing that the length of time considered is that during which the 
quantity, H, of heat is supplied from the first source, we find by integration 
H as 
ie Sah pat. 
But the value of g, when ¢=T, is the final remainder discharged into the refrigera- 
tor at the temperature T ; and therefore, if this be denoted by R, we have 
ey ae 
bog = 5 fp Hae ford ‘eilto 4! COL 
1 7s 
R=we7s fru pievultl. ol) ay Sn 
Now, the whole amount of work produced will be the mechanical equivalent of 
the quantity of heat lost; and, therefore, if this be denoted by W, we have 
WING = Re ee ee 
from which we deduce 

