716 
MR. A,. Me AULA Y ON THE MATHEMATICAL 
So far as it depends on p, l will be supposed only to contain the term — D m W, where 
D m is the density of matter in the standard position, and W an ordinary potential 
(quite independent, however, of electromagnetic phenomena). Of course, so far as 
it depends on p , l is supposed only to contain the term — D m p'*/ 2. Thus S 1 consists of 
the following parts :— 
- S SpV'l = DJ3 Sp'V'W .(1), 
— S Sp'pWl = — D,«S p Sp' — — d(D m Sp S p')/dt + D m Sp' S p . . (2), 
- S SD D VZ. (3), 
- S Sd d VZ.(4), 
- S SC c Vl = - S SD C VZ = - dS c Vl SD /dt + SS D d c Vl/dt . . . (5), 
- S SH E Vl = - SB SH/4n-.(6), 
— S Sd^G^ = wS Sp\(j>V\ .•.(7), 
where G stands, as throughout the present paper it will stand, for *G, and where </>' 
is defined by saying that 
<£ = - 2 ai .. ( 8 ), 
and that <f) is of Class I. in § 9 above. The proof of equation (7) is exactly parallel 
to the treatment of <£> in § 39 above, and, therefore, need not be given here. 
45. The part of SL due to (7) is 
jjjs &p\tfV\ds = - [jjsSp'^V'jA' + jjsSp'f 82' 
bv eq. (4), § 5, above. The part* due to (6) is [§ 26, eq. (19)] 
- (47T)" 1 (jjsVASHds = - (47T)" 1 (jjSAVSHefe - (Ttt)" 1 jj SA SH d$. 
When considering the whole of space this surface integral can be neglected, since 
by eq. (15), § 25, [V dSK\ a + b = 0, and by eq. (20), § 26, [Vd2A] a + 6 = 0. If, as for 
* This transformation which assumes a fact still to be proved (viz., that B = YYA, [Y d2A]„ + a = 0) 
is given, not wdth the object of determining the equations of motion, in which process this fact will not 
be assumed, but to find the rate of change of energy in an assigned space. 
