36 



Mr. C. Chree. 



t = x 2 f,c/2k, after which it steadily falls. The times at which the 

 increase and decrease are fastest are respectively the smaller and the 

 greater root of the quadratic equation 



^ rf 4** + V4* / ' (2) 

 and are approximately '0917^ ~, (3) 



and -908^ ... (4) 



4& * 



Supposing it were possible suddenly to supply a quantity of heat 

 to the surface of the liquid in the tub and to ensure that no com- 

 mensurable quantity was subsequently gained or lost, an observation 

 of the time at which the temperature at a given depth was rising 

 fastest, or was stationary, wo aid enable Jc to be determined at once. 



In the actual case the problem is more complex as /(%)> though 

 diminishing rapidly as % increases, is different from zero ; the 

 principle however is practically unchanged. By differentiation we 

 obtain from (1) 



2 / 7rk dh) 



pc dt 2 



+ terms at the limits (5) 



Now when t is moderately large the terms at the limits may be 

 neglected. This follows from a consideration either of the mathematical 

 form or the physical meaning of those terms. They are proportional, 

 one to the temperature instantaneously produced at the depth x at the 

 time t by the heat at that instant passing into the liquid from the 

 dish, and the other to the rate of change of this instantaneous effect. 

 Now even when heat is being very rapidly communicated to the 

 liquid, as at the commencement of the experiment, the rise in tem- 

 perature due to conduction at a moderate depth is for a minute or 

 two insignificant. Thus when the heat is being communicated very 

 slowly, as is the case at the time at which we shall employ (5), the 

 terms at the limits are for all practical purposes negligible. 



When the temperature of the liquid at depth x is rising most 

 rapidly, dvjdt is a maximum, and so d^vjdt 2 = 0. From the above 

 reasoning it follows that the time in question must satisfy the 

 equation — 







o 



+(*)>■ • '•«> 



