16 
PROFESSOE A. E. H. LOVE OX THE INTEGRATIOX OF THE 
Accordingly, the contribution of cto to the left-hand member of equation (31) is 
— d7r/oC"h 
In obtaining this result, we have interchanged the order of the integrcHion with 
I'espect to t and the integration over the surface cto. This step is certainly permissible 
if the subject of integration is, in each case, a continuous function of x, y, 2 ., t, for all 
the values that occur. Equation (35) and this condition can be satisfied in any 
numijer of ways, and, In particular, by taking a very large value of /r, and putting, 
after Kikchhoff, 
provided we supj)ose that is small of order 
With the same choice of (/)y, q, the rightdiand mendjer of eipiation (31) can have 
the limit zero. For this it is sullicient that, for all points between ctj and cto, the 
quantities v, w' and u, r, li' should be ultimately zero when t is either q or q. 
This is the case If /• -(- cq is negative and r -j- cq is positive for all values of r that 
occur. Equation (31) then takes the form 
II bS I (nih — ng) -)- v' {nf ~ Ih) + w' [Ig — iitf) 
[iriLV — riv) + g'{itif — br) -|- h'{h' — niu)'^ = 47r/y‘“^ . . (37), 
where tiie surface integration is taken over cr^ only. 
The (juantities u\ o', iv' and /', g , h' liave values, which are not extremely near to 
zero, only when 0 -j- is very near to zero, r l)eing tlie distance of a ])oint on the 
surface from the point (J. The integration with respect to /, in the left-hand member 
<.)f (37), can accordingly Ije carried out by observing the rules 
X<fio^lt = 
1 
c 
(xX^- 
c 
^0 
v thus find for tlie value of/i at the point (), and at the time t = 0, the 
iM j\ialion 
