14 
PKOFESSOR A. E. H. LOVE ON. THE INTEGRATION OF THE 
We integrate this equation with respect to the time, between two fixed values q, 
and q, and we thus obtain the equation 
[ ilf [ — h'm) + V {111 — f'n) + v:{ f'm — rfl) 
-\- f{v'n — w'ra) + g{w'l — u'n) + // {um — v'l)]d^ 
{nu — ii'u + vd — vv + dir' — v''iv)dT .( 31 )- 
J'o 
If tlie functions involved are sucli tliat tlie volume integral on the right vanishes, 
and that the order of integrations on the left can he interclianged, we have 
I c/S j — h'm) + v{k'I — f'n) + iv{f'm — g'l)]dt 
^0 
= ~ ~ + w{fm — gl)}dt . . . (32). 
Tills equation plays the same part in the present theory as Green’s ecpiation 
^ ci^ i] cv 
Y and ({) being harmonic, plays in the Theory of Potential. 
Integration of the Genered. Equations. 
15. We shall now siq^pose the boundaries of the region of sjoace, to which the 
theorem of § 14 is applied, to be (1) a closed surface o-j containing none of the 
singularities of the functions u, v, rv, (2) a small sphere cr., with its centre at a 
point O, inside cr^. Then, taking O as origin, we shall assume for u', v', iv, the 
expressions (17) of § 11, and for y', g', h' the corresponding expressions (18). For 
/, m, n at ixo we have to put ic/r, y/r, z/r, and the contribution of cro to the left-hand 
member of (3l) becomes 
We take to be of the form ^(r -j- cV), so that 
(84), 
1 c)0 
dr C df ’ 
