212 ME. W. F. G. SWANN ON THE SPECIFIC HEATS OF AIR AND 



the flow. Let C, E, J, S, Q have the significance assigned to them on p. 200. Let 

 80 be the rise in temperature in a length S.r of the calorimeter, 6 the excess of the 

 temperature of a point in the calorimeter over that of the jacket, r the resistance of 

 that length of the heating coil which occupies 1 cm. of the axis of the calorimeter, 

 a the temperature coefficient of the material of the heating coil, f the emissivity of 

 the surface of the calorimeter in watts per square centimetre, and p the perimeter of 

 the calorimeter proper. The differential equation which applies after the steady state 

 of temperature has been attained in the calorimeter is 



JSQS0 = CV,, (1 +aO) 8x-f 2 )08j.; 

 or 



(1). 

 For brevity, write 



A = 0$,-C 8 r )/JSQ, B = CV /JSQ, y=^>/JSQ - - - (2). 



Taking the, origin of ,<: at the point where the gas first meets the heating coil, the 

 solution of (l), subject to the condition = when ,/: = 0, is 



= (L-r- Al )B/A. 



Since A is a quantity of the first order, this equation may be expressed in the form 



= Bx'(l-iA.x) .......... (3), 



the equation applying throughout the length of the heating coil. The temperature 

 h at the point where the gas leaves the heating coil, is obtained by putting x = I in 

 this equation, I being the length of the axis of the heating coil. 



After the gas has passed the heating coil, the differential equation (1) reduces to 



= 0. 

 The solution, subject to the condition that the temperature is O t at .c = /, is 



8 = B/(l-JAZ)e- (jr -", 

 or to the first order 



(x-i)} ........ (4 



The temperature rise , measured by the thermometer H, is the mean value of 

 over its length, thus 



]} ...... (5), 



where m and n are the values of x corresponding to the two ends of thermometer H. 

 The true mean temperature over the path, from the beginning of the heating coil to 

 the end of thermometer H, is 



6 m = - \\'-Bx(l-$Ax) dx+ [El (1-p.Z) [1-y (x-l)]dx\. 



Tl I J o J i 



