Mr. W. J. M. Rankine on Heat as the Equivalent of Work, 103 



tion of the entire spheroid, from the gradual cooling of all its 

 particles, would lessen its dimensions, and thus tend to accelerate 

 the velocity of rotation so as to diminish the length of the day. 

 The energy of both of these opposing tendencies depends upon 

 a common cause, the rate of cooling of the entire earth. This 

 has been demonstrated by Fourier and other illustrious mathe- 

 maticians, on the most favourable suppositions to rapidity of 

 refrigeration, to be so extremely slow, that if only one of the 

 counteracting influences here adduced existed without the other, 

 we could scarcely expect to discover its action on the rotation of 

 the earth until after a long period of exact observations. When 

 the fact of the simultaneous existence and opposition of these 

 influences is remembered, it should not excite surprise that astro- 

 nomical observations should have hitherto never disclosed any 

 variation in the length of the day, and ages may possibly elapse 

 before any such variation will be discovered. 



XII. On Heat as the Equivalent of Work. 

 By W. J. Macquorn Eankine, C.E., KKSS.L, ^ E. 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 



IN the Number of your Magazine for July (at p. 75) there 

 appears an abstract of a paper by M. Hoppe, first published 

 in Poggendorff's Annalen, vol. xcvii. p. 30, commencing, " Hoppe 

 has contributed a memoir upon this most interesting and im- 

 portant subject, which places the analytical theory in a remark- 

 ably clear, simple, and general point of view, so far at least as it 

 relates to permanent gases.^^ This observation naturally leads 

 the reader to infer, that the theory previous to the publication 

 of M. Hoppers paper was deficient in clearness, simpHcity, and 

 generality ; and on first reading it in Silliman's Journal, I con- 

 templated entering into a detailed discussion of M. Hoppe's 

 paper; but this intention I have since abandoned, on finding 

 that that discussion has been already made by Professor Clausius 

 (Poggendorff^s Annalen, yo\. xcvii. p. 173) in a short paper, which 

 appears to me to be well worthy of publication in the English 

 language. I shall therefore confine my remarks to stating, that 

 the whole theory of heat as the equivalent of work performed by 

 the expansion of elastic substances, whether gaseous, liquid, or 

 solid, is summed up in this one equation : — 



Jdq=Udr + r{dr^+dv.^)ff^dv; . . (A) 



where J is Joule's equivalent ; dq the quantity of heat received 

 by the substance during the increase of absolute temperature dr 



