692 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
conclusion is that V', V" being the values at points within and without the bounding 
surface, V' and V" are in general different functions of the coordinates (a ... c, e) of 
the attracting point; but that at the surface we have not only V'=V", but that the 
first derived functions are also equal, viz, that we have 
dV'__dW dV' d\" dV dV" 
da da ’ dc dc 5 de de 
27. In the general case of a Potential, 
T7 p dx . . . dz dw 
V {{a-xf . . . + {c-zY + (e-wfY a ~ i ’ 
if g does not vanish at the attracted point (a ... c, e), but has there a value g' 
different from zero, we may consider the attracting (s-fl)dimensional mass as made 
up of an indefinitely small sphere, radius s and density g', which includes within it the 
attracted point, and of a remaining portion external to the attracted point. Writing 
V to denote^ • • • +^ 2 +^ 2 ’ then, as regards the potential of the sphere, we have 
VV= — 
4 (Hr) 
r(**- 
T) ? 
(see Annex III. No. 67), and as regards the remaining portion 
VV=0 ; hence, as regards the whole attracting mass, VV has the first-mentioned value, 
that is we have 
(- 
\da 2 
d 2 
1 A 1 Y \ V— _ 4 (IA) S , 
da 2 ‘ ‘ ' ' dc 2 ' de 2 ) P(i s— i) ^ 5 
where g* is the same function of the coordinates (a ... c, e) that g is of (x ... z, w) ; 
viz. the potential of an attracting mass distributed not on a surface, but over a portion 
of space, does not satisfy the potential equation 
( d 2 . d 2 d 2 \~. r „ 
d#- • • +^+^y v=0 ’ 
but it satisfies the foregoing equation, which only agrees with the potential equation 
in regard to a point (a. . .c,e) outside the material space, and for which, therefore, 
g' is =0. 
The equation may be written 
S'= 
r (js-j) / d* 
4 (Pi ) s+1 V ® 2 ‘ ' ’ 
+ dc 2 ^de 2 > V ’ 
or, considering V as a given function of (a ... c, e), in general a discontinuous function 
but subject to certain conditions as afterwards mentioned, and taking W the same 
function of {x ... z, w) that V is of {a ... c, e), then we have 
S= 
r(-b-j) 
4(ri) s+1 
. . (D) 
