Whence we have d 



206 Mr. Ilerapath on True Temperature, and the [Sept. 



bodies. Moreover the Fahrenheit arithmetical mean is equal to 

 t+£ .JV + 5 ^ Hencel^-^-flZ-^-^Ms 

 the distance of one result from Fahrenheit's arithmetical mean, 



(T P + n T P \- T a + T J 

 'p n p 'j ^ — the distance of the other ; and, 



.. f /T,P + nTP,\2 ; /TP + nT.P.x* ,„,. ,'-*«. , 



therefore, (- TTTp7 -J + ( g + » P , ) ~ 0? + T » the 



difference of these distances, which, to satisfy the object of the 

 problem, must be a maximum. But the temperatures T, T t , and 

 the particles, P, P,, being the same, n is the only variable. 



\ P + » P, / V P + ii p, j 



T, P + nTP, ,/ T,P + wTP, \ _ T P + n T, P, , /TP+nT,P,\ 



P + n P, ( P + i»P, J = : P + ii P, I P+ iiT77' 



But omitting; d n, which is common to both sides, d — !— — ■ 



° P + nP, 



T P, (P + n P,~) P, {T, P + » T P,~> _ TP,P + nT P, : -T,P,P-nTP 1 a 

 (P + n P,) 4 ^P + "P,) a "(P + " Pi)* 



_ (T - T.) P P, , , T P + n T, P, __ T,P,(P + n P.) _ 



(P+nP,) 2 ' P + nP, (P+nP,) 5 



P, rT P + ii T, P.) __ T, P, P + n T, P, 2 - T P, P- n T, P," _ (T,-T).PP, 

 (P + n P,) 1 " (P + n P,) 2 _ (P + n P^ " 



The two differential parts, therefore, in the above differential 

 equation being the same quantities with contrary signs, and the 

 sides of the equation itself having contrary signs, we have 



T, P+«TP, T P + n T, P, , ., (T,-T).P 



-pTITpT = p + ,p, i and consequently n = [^pr^-jr 



P w r 



= - . Restoring the value of n, that is — ' we get P l W l = PW > 



and also N, V, = N V. Consequently the most advantageous 

 method of examining the theory is when the interval of the tem- 

 peratures is the greatest possible, and the weights reciprocally 

 proportional to the particles in equal weights, or the volumes 

 reciprocally proportional to the numeratcms. That is, the tem- 

 peratures being the same, the best method to make the experi- 

 ment, or to examine the theory, is when the weights of the fluids 

 are so related to the particles in equal weights, or the volumes to 

 the numeratoms, that t = -*• (T + T,), or that the temperatures 

 are symmetrical with respect to themselves and the fluids. 



Cor 1.— When P, = P or N, = N, W, = W and V, = V ; 

 so that in parts of the same fluid the most advantageous circum- 

 stance under which the theory can be examined, is when the 

 portions mixed are equal \ which coincides with what we have 

 deduced in Prop. II. 



It is proper to remark, that in this deduction one of the dist- 

 ances from the Fahrenheit arithmetical mean becomes negative 

 with respect to the other; so that the true difference of distances 



