20 
PEOFESSOR A. E. H. LOVE ON THE INTEGRATION OF THE 
The integral last written is 
of which the volume integral vanishes identically. The terms which contain ii, v, w 
exj^licitly, and those which contain F, G, H explicitly, may be transformed in the 
same way, and we have finally 
u = ||wqC/S + llwy/S + II c/S j ; . . . . (39); 
and this is identical with Poisson’s integral’^ of the ecpiation 
in terms of initial conditions. 
18 . The results can be interpreted in terms of sources of disturbance of the two 
types previously investigated. Any point of the surface S must be regarded as the 
seat of a source of the first type, and of a source of the second type. The axis of 
the source of the first type is at right angles to the direction of (F, G, H), and is 
tangential to the surface ; its radiation function is the product of c/S, the resultant 
of (F, G, H) and the sine of the angle, which the direction of this resultant makes 
with the normal to the surface. The source of the first type is equivalent to three 
sources, with their axes parallel to the coordinate axes, and with radiation functions 
equal to 
- c/S(mH - nG), - dS{nF - /H), - dS{/G - mF) ; 
these expressions, for any point on the surface, are functions of t, and they take the 
characteristic form of radiation functions, when t — r/c is sidistituted for t. 
Tlie axis of the source of the second type is at right angles to the direction of 
{u, V, w), and is tangential to the surface ; its radiation function is the product of dS, 
the resultant of {u, v, w) and the sine of the angle, which the direction of this 
resultant makes with the normal to the surface. The' source of the second type is 
equivalent to three sources, with tlieir axes parallel to the coordinate axes, and with 
radiation fimctions eijual to 
— dFi{'mw — nu), — dd){iiii — Iw), — dS{lv — mu); 
* The form of Poisson’s integral usually given requires the peitonuance of differentiation, with respect 
to f, upon an integral taken over a sphere of radius C/, and thus, when the differeutiation is carried out, 
there will he three terms in the com[)lete expression; it is easy to verify that these terms are precisely 
those given in equation (39). 
