1 hermometers ivitli Spherical and Cylindrical Bulbs. 139 



Following a method of Riemann's, let us consider the 

 function p^(r, t) defined by 



^ 2JJai rlc)e c'W, 



This function satisfies the differential equation (7) and 

 vanishes for t = 0. Further, 



X (c 3 = 0, t < ; 



Xfc, = 1, * > 0. 



Take a series of small intervals f = t , Y — t . defined by 

 the fixed values of r, 



0, t u t 2 , . . . , t p , 



JS r ow consider 



«i = 2 <Kt p )[.x(r,t-t p ) - x ( r ,t-t p+i )]. 



We see that when r = c, 



"i = <t>(* p ) *<>r *„ < *< ^ +r 



Hence in the limit, as f->0, we get, since U\ satisfies (7), 

 and vanishes for t = 0, and in addition is equal to <f>(t) at 

 time t, when r=c, 



Performing the differentiation, we have the following 

 solution lor our problem : 



C* " 2J (« r/c) a 2 af t <*% , 



° nil Jl K) «V 



Take <p(t)=Gt, where Gr is constant. Then we find, on 

 integrating, 



u - 2G 2, — i7-r '+ti « f — 1) 



. (9) 



