186 Mr EL. W. Hobson, On a Radiation Problem. [Feb. 27, 
where w, denotes what w becomes when «+2 is written for x; the 
required expressions for v are then 
a hf (eto 
I=). [ Sas FREE’ .e 4(t-aA) dzdn...... (4) 
Pd Osean F (w+) 
— = @?—hz = 
ah foe q fy ies | dedg idence (5). 
2qV/« 
As in the case above mentioned in which the temperature at 
x =( is given, the rod may be replaced by a rod, infinite in both 
directions, with a doublet at the origin, so in the present case the 
rod may be supposed to be continued on the negative side of the 
origin with a distribution of doublets along the whole of the part 
of the rod so continued; in fact the equation (4) shews that if, in 
the infinite rod, doublets be placed of magnitude 2hke-’*.dz. f(8) 
at the element dz which is at the pomt # =— z, the temperature 
at any point on the positive side of the origin due to all this 
continuous distribution of doublets will be the actual temperature 
of the given semi-infinite rod under the given radiation condition, 
supposing the initial temperature of the rod to be zero. The tem- 
perature due to the doublet at «= —z is 
t e+2)2 
| Dy dz ae 2) ‘ eo“! e-ie-¥) f (d) dx, 
o2 Jae (t—r)? 
and the complete temperature is that obtained by summing this 
expression for all values of z from 0 to «. Hence corresponding 
to Thomson’s single doublet at the origin, there must be in the 
radiation problem a distribution of doublets on the whole of the 
negative part of the rod, and their magnitude diminishes in geo- 
metrical progression with the distance from the origin. 
Hitherto it has been supposed that the initial temperature 
over the semi-infinite rod was zero, but the formula (3) may be 
applied to obtain the expression which must be added to the 
expression (4) or (5) when the initial temperature is arbitrarily 
given, say, ¢(a#). In this case w must be determined from the 
conditions w= (1 _ : +) ¢ (x), when t=0 for all positive values 
of w, and w=0 when ¢=0; now the expression 
1 ih _(@-2!? a ttl nding 
— e@ det —@ dnt (w')dx 
aN Kt 0 v 
is the temperature due to a series of sources of magnitude ¥() dx 
on the positive side of the origin and of sinks of magnitude 
