702 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
Iii exactly the same way if (a ... c, e) be an exterior point, then we have 
is-fPJv ; 
and adding, and omitting the terms which vanish, 
that is, 
4(ri) s+1 v „ 
r(i*-i) v ’ 
-r(is-i) /dw' dw" 
4(riy +1 v d8 ' d8 " 
dS 
(a— x) 2 ...(c — z) 2 + (e—w) 2 y s ~ s ' 
42. Comparing the two results with 
§ dS 
(a—x ) 2 . . . + (c— z) 2 + (e— w) 2 } lS ~*’ 
we see that V', V" satisfying the foregoing conditions, there exists a distribution § on 
the surface, producing the potentials V' and V" at an interior point and an exterior 
point respectively ; the value of g in fact being 
§=- 
TQs-i) id W' dW \ 
4(ri) s+1 V da' + dx" )’ 
(C) 
where W', W" are respectively the same functions of (x ... z, w) that V', V" are of 
(«...<?, e). 
The Potential-solid Theorem D. — Art. No. 43. 
43. We have as before (No. 40), 
jw ^ dS-t- jit . . . dzdw WVU- 
= J u w ds + f<& ■ • • * trvw-ipA^ T, 
where, assuming first that W is not infinite for any point (x ... z, w) whatever, we have 
no term in T ; and taking next U = — — — ■ — — - — - — 7 , ot i._i as before, we have 
& {(a-x) 2 . .. + (c-zf + (e-w) 2 y s 2 
VU=0 ; the equation thus becomes 
J W W dS ~ j U 7F V =f* * dw UVW ^ 
where W may be a discontinuous function of the coordinates (x . . . z, w), provided only 
there is no abrupt change in the value either of W or of any of its first derived functions 
^ viz. it may be any function which can represent the potential of a solid 
mass on an attracted point (x ... z,w); the resulting value of V W is of course discon- 
