428 Prof. Clausius on the Application of the Mechanical 



This equation may be written in the following form : — 



T,^«=C+a(/,-/8)-*(;>i-M . . . (47) 

 wherein the magnitudes C, a, and b are independent of t^, and 

 have the following values : — 



C 1 [>•■ i,vf '-o-^(T.-To) \1] 



b = 



AkieY-la) 



(47fl) 



Of the three terms on the right-hand side of (47), the first far 

 exceeds the others ; hence it will be possible, by successive ap- 

 proximation, to determine the product T^ff^, and thence also the 

 temperature t^- 



In order to obtain the first approximate value of the product, 

 which we will call Ty, let us on the right side of (47) set t^ in 

 the place of t^, and correspondingly jOj in place of jOg* then 



Ty=C (48) 



The temperature /', corresponding to this value of the product, 

 can be sought in the table. In order to find a second approxi- 

 mate value of the product, the value of i' just found, and the 

 corresponding value of the pressure p', are introduced into (47) 

 in the places of t^ and p^j whereby, having regard to the former 

 equation, we have 



Ty=Ty + «(^i4-0-^(Pi-y). .-. . (48«) 

 As before, the temperature /", corresponding to this value of the 

 product, is given by the table. If this does not with sufficient 

 exactitude represent the required temperature t^, the same method 

 must be repeated. The newly-found values t" and p" must be 

 substituted in (47) in place of /g and p^y whereby with the assist- 

 ance of the two last equations, we have 



T'yf^TV + a{t'-t")-b(p'^p% . . (48^) 



and in the table we can find the new temperature i'". 



We might proceed in this manner for any length of time, 

 though we shall find that the third approximation is already 

 within jl^dth, and the fourth within xoVzj^^th of a degree of the 

 true value of the temperature /g. 



48. The treatment of the third of the equations (XVII) is 

 precisely similar. If we divide by Y—la, and for facility of cal- 



