1888.] Mr E. W. Hobson, On a Radiation Problem. 185 
in which « is the conductivity of the rod divided by its density 
and specific heat, represents the temperature at any point of the 
rod when the initial temperature is zero gue the end #=0 is 
maintained at temperature f(t); writing ——_— ee this formula 
G= wae 
becomes 
o F e 
Me = e-1 a Ie (2). 
au 
Sir W. Thomson has shewn (‘ On the Electric Telegraph,’ Col- 
lected Works, Vol. 2, No. Lxx11.) that these formulae may be ob- 
tained by supposing a doublet consisting of a source and sink to be 
placed at the point «= 0, the rod being supposed to be continued 
on the negative side of the origin; the product of the amount of 
heat generated in time dt by the source, into the distance between 
the source and sink, is equal to 2«f(¢)dt; the formulae (1) and (2) 
represent the temperature due to this doublet. 
In the present communication formulae are given corresponding 
to (1) and (2) for the case in which instead of the temperature at 
the plane = 0 being given, radiation takes place across that plane 
into a medium of which the temperature is given, say f(t). In 
this case the boundary condition which holds when #=0 is 
du 
# hiv (0), 
where h denotes the external conductivity. Let w=v— = , then 
ad =) ate 5G 1) me fi 
= © x w= ( he) pee dey 
thus wu satisfies the equation of conduction and can be determined 
from the conditions that u=0 when ¢=0 and w=/(¢) when #=0; 
in fact wu is given by either of the expressions (1) and (2). 
d al —hz 
We have dn ) =— hue~™, 
hence v= he | ue" dé, 
x 
the upper limit is infinite because v must vanish when w is infinite; 
let €=+ z, then the expression for v becomes 
ir (Dae Tchad ep ea ae ee (3, 
