246 Professors W. E. Ayrton and J. Perry on 



where v is the excess of temperature of the point over the ex- 

 ternal temperature on the Centigrade scale; 

 v is the excess of the initial temperature over the ex- 

 ternal temperature; 

 t is the time in seconds since the ball began to cool; 

 t is the radius of the ball, in centimetres; 

 K is the internal conductivity, in centimetre-gramme- 

 second units; 

 E is the surface emissivity, which is such that from a 

 square centimetre of surface 1 Centigrade degree 

 higher than the external medium there will be emitted 

 per second a quantity of heat E; 

 C is the specific heat of the substance per unit volume; 

 a is an angle such that 



1 _& = . 



Iv tan a. v J 



from which equation successive values of a must be found and 

 used to form the different terms under the sign of summation 

 in equation (1). It is evident that the values of a will lie 

 successively in alternate quadrants — the first, third, fifth, &c, 

 or the second, fourth, sixth, &c, — also that after a certain time 

 has elapsed, depending on the values of E, K, and r, all terms 

 after the first become negligible, so that the true curve of v 

 for a given point becomes a simple logarithmic curve. 



If the point is at the centre of the ball, then, as x is nought, 



X ' 



a — 

 r 



and the general equation becomes 



2Er^ sin a «« /Q . 



v = v —-X— . ■ € c,-2. ... (3) 



K / ., sin 2c * 



a (i-^) 



or, when all the terms except the first become negligible and 

 therefore a. is the smallest positive angle satisfying the equa- 

 tion (2), we have 



2Er sin a -£« '.. 



r = r °K- A sin2«\ 6 " ' ' ' • W 



Equation (4) may also be written in the forms 



sin a — a cos a _ °^ Kt /rs 



V=2V ; 6 Cr2 (5) 



a — Sill a cos a v 



