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empirical formulas, jtnblislioil in !iis woi-k on tlio theory of the .team- 

 engine, in 18-14, for the nhitiou of vohmio i;nd pressure during expan- 

 sion, and liad based his (ltterraiuatioi:s of the quantity of work done, and 

 of cxponditu' J of steam in the engine, upon this set of assumptions and 

 formulas, considering tLe .-^teaui to remain in its initial condition of dry 

 and sacurated vapou'-, or of moist vapour as the case may be, from the 

 beginning to the end of th'> stroke. Eirurs were thus introduced, which, 

 altliough not important in companion witli those often occurring when 

 the result of jjurcly thermodynamic and in so far Lorrect treatment 

 was compared with the actual case, were, nevertheless, .sufficiently great 

 to become noticeable when the true theory of heat-engines became known 

 and cori'ectly applied. Ciausins proved that, in the exprinsion of dry and 

 .saturated steam tloing work in the engine, eondensatlon must take place 

 to a certain extent, and that conso(|ucntl3- the weight of steam in the 

 cylinder must bo somewhat reduced by the process of exj)ansion beyond 

 the point of ' cut-off.' During the period of compression, also, the reverse 

 effect must occur, and the compressed mass must become superheated, 

 if initially dry. He showed that the amount of work actually dono in a 

 non-conducting working cylinder must be sensibly different from that 

 estimated bv the method of De Pambour. Taking advantajro of the 

 re-determination of the constants in llegnault's equations ejected by 

 ]\loritz, Clausius obtains immerical results in the application of the 

 true theory, and deduces the amount of work done in the steam-engine 

 under various conditions such as are met with in practice. He .shows 

 how the action of the engine may bo made that of the Carnot Cycle, and 

 determines the effect of variation of the temperature of the 'prime" 

 steam. The investigation is, in the main, purely theoretical : no appli- 

 cation is made to the cases met with in real work, and the comparison 

 of the results of the application of the new theory to i)ractice in steam 

 engineering is left for others. 



The work of Clausius is, throughout, perfectly logical, and beautifully 

 simple and concise, and his application of the theory to the steam-engine 

 amounts to a complete reconstruction of the work of Carnot and his 

 followers upon a correct basis. He develops with mathematical exact- 

 ness of method and work the fundamental principles of the science of 

 thermodynamics, constructs the 'fundamental equations,' the so-called 

 ' general equations of thermodynamics,' and, in the course of his work, 

 proves the fact of the partial condensation of saturated steam, when per- 

 mitted to expand, doing work against resistance. 



Professor llankine began his work upon tlie theory of the transforma- 

 tion of heat into mechanical energy at about the same time with Clausius 

 (1849), and published hiis first important deduction, the form of the 

 general equation of thermodynamics, nearly simultaneously, but a little 

 earlier. He gave mnch attention to the then incomplete work of develop- 

 ment of applied thernnjdynamics, and produced, not only the whole theory 

 of the science, but very extended papers, including solutions of practical 

 problems in the application of the science to heat-engines. Stating Avith 

 singular brevity and clearness the main principles, and developing the 

 general equations in substantially the same form, but by less-easily- 

 followed processes than his contera])orary, he proceeded at once to tlieir 

 application. He determines the thermodynamic functions for air and 

 otlier gases, exhibits the theory of the liot-air engine, as applied to the 

 more important and typical forms, deduces expressions for their efficiency, 



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