6 
rROFESSOE A. E. H. LOVE ON THE INTEGRATION OF THE 
(3) </) satisfies the equation 
c^4>/ct~ = C~ V ~<j) .(l), 
at all points outside S, C being a definite constant, and denoting the operator 
‘Vo 
o- ^ 
OX- oy- cz- 
The value of (h at any point O outside S, and at time t, can be expressed as a 
surface-integral taken over S ; in fact this value is 
0(f) 
'V 
CV 
+ 
1 dr 
Cr du 
r^rk 1 
ccf) 
dt 
.(^)) 
where dv denotes the element of the normal to dS drawn outwards {i.e., into the 
region of space where 0 is situated), and the expressions in square brackets [] are to 
be formed, for each point of the surface, at the time t — the, r being the distance of 
the point of the surface from the point 0. 
This theorem was obtained by Kirchhoff"^ by an application of Green’s theoremt 
to the function ^ and an auxiliary function V, which satisfies the equation (l) at all 
points except O, and has the form 
T {r + CQ 
’ .wb 
where F is a function to which suitable properties are assigned. 
8. [Partly re-written March, 1901.]—If (f) were the velocity potential of sound 
waves in air, the terms of (2) that contain [<^] and [d(j)/di'[\ would be interpretable in 
terms of velocity, and those that contain [d(f)/dt'] would be interpretable in terms of 
condensation ; the expression (2) would represent the motion at any point as due to 
sources of definite types distributed over the surface S. But, if ^ is one of the 
components of a vector quantity, 23ropagated by transverse waves, [0^/0^’] has no 
physical significance, and the expression (2) cannot be interpreted in terms of appro¬ 
priate sources of disturbance. 
Again, the expression (2) may be interpreted as showing that every element of the 
surface S becomes the centre of diveiging secondary waves. If (f) is one of the com¬ 
ponents of a vector quantity, and the primary waves are transverse, the application 
of Kirchhoff’s theorem is open to the criticism that the secondary waves are not always 
‘Vorlesungen ii. math. Optik,’ pp. 23-27. 
t The theorem referred to is the one expressed hy the equation 
