On the Tension and the Latent Heat of different Vapours. 483 



Thus the solution of the proposed is made to depend upon those 

 of equations of a lower order. 



Examples might be given to illustrate the use of the theorems 

 (B) and (D), which are a generalization of (A) and (C) ; but the 

 method must be plain from what has been done, the only dif- 

 ference being that we should have <f){x) in the room of Xj and 

 ^(D) in the room of D, and the symbols tt and m would be more 

 general. 



In reducing equations to the forms {!), (2), &c., we shall 

 obtain from the same equation results of different forms by giving 

 different forms to the arbitrary functions \[x), X(D) ; which is 

 one advantage which these functions give us, and we must give 

 them that form which will render our transformed equation the 

 most convenient for solution. But in some cases, instead of sup- 

 posing X(^), X(D) integer, or even rational functions, of x and 

 D, we may so determine their form as to take away from the 

 equation functions of them which are neither integer nor rational. 



The examples (5) and (6), with their solutions, might be de- 

 duced from those of (3) and (4) by the commutation of symbols 

 mentioned further back, and by a suitable change in the func- 

 tions concerned ; and it may be presumed universally, that by 

 making this interchange of symbols in any linear equation and 

 its solution, we shall obtain as the result another equation and 

 its solution. But when the coefficients of a differential equation 

 are integer functions of x, those of the commuted one will like- 

 wise be integer functions of Xj and in this respect they will be 

 alike. Therefore either of the formulae (A) and (C) may be em- 

 ployed to effect its solution, but not perhaps with equal facility 

 or equal success. A variety of means, however, is better than 

 one only, as it augments our chances. 



When the second member of an equation contains three or 

 more terms, its solution may sometimes be made to depend on 

 the solution of several other equations having only two terms in 

 their second members; but I cannot enter upon that subject 

 here. 



October 30, 1851. 



LXXII. On the Theoretic Connexion of two Empirical Laws re- 

 lating to the Tension and the Latent Heat of different Vapours. 

 By 6. Clausius*. 



A SUPERFICIAL contemplation of the tension series, expe- 



^•^ rimentally developed for the vapours of different fluids, 



suffices to show that a certain uniformity exists therein ; and 



* From PoggendorfF's Annalen, vol. Ixxxii. p. 2/4. 



