1888.] Mr E. W. Hobson, On a Radiation Problem. 187 
— pv («) dx on the negative side of the origin; it therefore satisfies 
the equation of conduction, vanishes when «=0, and is equal to 
(xz) when t=0 and ~ is positive; putting W(x) =¢(#) — : p (2), 
we have therefore the value of u, hence the required value of v is 
from (3), 
h ri - eet _(e@te+a’/)2 pas! An 
————— — he K = Ant ae AN gl fi 
—— an et ote Hoe) 7 8@) dede! (6). 
The sum of the expressions (4) and (6) is the complete value 
of v which is initially equal to ¢(x) when «w is positive, and which 
satisfies the condition = =h|v—f(t)] when «=0. 
The expression (6) shews that the effect of the initial tem- 
perature ¢(a') over an element dz’ may be thus represented; sup- 
pose the rod extended to infinity on the negative side of the origin 
and suppose an infinite series of instantaneous sources stretching 
from the point #=a' to v= — ; the strength of the one between 
the points z and z+ dz being he-”dz | (a') — : p )} dx’, z being 
the distance of a source from the point «= 2’; and also suppose a 
series of sinks stretching from w=—w to w=—o of the same 
strength at the distance z from this latter point as the source 
at the same distance from w# = 2’; such a distribution of sources and 
sinks will produce a temperature which satisfies the given con- 
dition at the point 7=0 and has the given initial value. 
It has thus been shewn that the complete solution of the 
radiation problem, which is given by the sum of the expressions 
in (5) and (6), is a solution representing the temperature in a 
rod indefinitely extended in both directions due to a system of 
doublets, sources and sinks, the magnitudes of which are completely 
determined. 
The Fourier solutions may of course be deduced from the 
definite integrals in (4) and (6). 
(2) Notes on Conjugate Functions and Equipotential Curves. 
By J. Brix, M.A., St John’s College. 
1, Suppose that we have a certain function of a complex 
variable, w=f(z) = & + mm, then it follows that 
NG See re am 
i ace * Oy?’ 
VOL. Vi. PT. tv, 14 
