200 Mr. Haycraft on the Specific Heat of the Gases. 



In general it appears, too, from ideas that have occurred to 

 me, of which I may say more hereafter, that any equation, as 



F{x, tyx, i<Mr, 4>x 2 x, . . . ^* n x] = 

 has n arbitrary functions, provided however that a r x = x and 

 r >m ; and that an equation of the form 



F[.r, fyx, ty^x, .... ^"x] =- 

 has the same number. My ideas, however, on this subject 

 are not sufficiently matured to enable me to speak confidently. 



The method of substitution pursued in our solution of (1) 

 which is taken from Mr. Babbage's solution of \J/.r = <J/ xx, ap- 

 plies very successfully even in cases where it is introduced but 

 partially. Thus in the equation 



1 +fxhl>x + ^ux^=^x.^/ccx 8) 



which Mr. Babbage has solved by a very neat process of dif- 

 ferentiation and integration, but ultimately bringing out a 

 transcendental solution, it may be applied partially to the evo- 

 lution of an algebraic solution. By changing x into otx, we 

 find that fx =fccx when o?x = x, which is the condition of 

 the problem. For tyx on the left side only of the equation 

 substitute vtyx + b$x, and equate the result with the same 

 result, having x changed into ax; and we shall find 



<l>a.x =4>x + $x — <pax, 

 which substituted for ■i/a.x in (8) gives ultimately 



^x=\ K/{<px-focxf + 4(/t* + 1) +/r- tfrx-teur) 

 for the complete solution of (8). 



Had we substituted for tyx on the right side only of (8), we 

 should have found 



<ba.x = .\I/ x 



and therefore 



+*-*/Sf + (£-+»)'•>* + £(t=- + 



which is a solution equally general, but not so simple, as the 

 preceding. 



XXXV. On the Specific Heat of the Gases. By W. T. Hay- 

 craft, Esq.* 



THE experiments which I now submit to the Royal Society 

 are repetitions of those I made many months ago, for 

 the purpose of ascertaining the specific heats of the gases. 



* From the Transactions of the Royal Society of Edinburgh, vol. x. 



The 





