22 
PROFESSOR A. E. H. LOVE ON THE INTEGRATION OF THE 
the conditions (1), (2) and (4), and either (a, y) or (/, g, h) would he normal to S, 
at every point on S ; so that we should have either 
a : ^ : y — I : m : n, 
or f: g : ]i = I : rn : n, 
(/, m, n) being the direction cosines of the normal to S drawn inwards. 
In both cases 
I ^/t[[rhS[/(/3/^ - yg) + rii{yf — ah) + n{ag - f3f)] = 0. 
Now this integral is 
— t a 
dh 
the integral being taken through the volume within S, and this is 
(It 
+ r + ir) + + r) 
since a, y and /’ g, h vanish when t — The expression last obtained cannot 
vanisli, unless a, /3, y and f, g, li vanish, at all times, and at all points within S. This 
proves the theorem. 
It follows from this theorem that, if either the tangential components of (u, r, uj), or 
those of (y, f/, h), are given at all points of S, the solution of equations (8) and (9) 
of § 9 is unique. 
21. Now let (?q, ?f’j) and (7q, r.., v\^ 1)e two sets of possilde magnetic displace¬ 
ments, for wliicli tliere are no singularities within a closed surface S ; and let Fj . . . 
be the correspoiiding vector potentials, and g\ . . . the corresp(uiding electric dis¬ 
placements. Siqjpose further that, at a certain time all these vanish at all points 
within S. We shall apjdy the theorem of ^ 14 to tliese vectors. We identify the 
set with suffix 2 with the set previously accented (§ 15), S(7 that = ?f, . . . , the 
point (), which is the sde singidarity of tlie accented set, being outside S. Then the 
right-hand nu‘ml )er of equati(m (31) vanishes, and we'have 
U) 
h'rn) -f — /'ll) + n'/f m — /!)] 
hgn) d- v'{hj — fgt) + n/fgn — gj)]. 
If there is a set of magnetic and electric displacements, free from singularities 
within S, and making tlie tangential components of the vector potential on S the 
