684 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
N, we have then 
d© 
V ~~di n ~ dP 
,yi 6?© <?© 
1 ~W’ ~ ~ds' 
dW' dW' dW" dW 
We have in like manner = — -y- ; and the equations obtained above 
may be written 
dW' _d® 2(ri) s+1 
ds ~ ds ~ r(is-i) St 
dw"_^© 2(ri) s+i 
ds ~ &+r(i* -*)£»■ 
in which form they show that as the attracted point passes through the surface from 
the position P' on the one side to P" on the other, there is an abrupt change in the 
dW dV 
value of or say of the first derived function of the potential in regard to the 
orthotomic arc s, that is in the rate of increase of V in the passage of the attracted 
point normally to the surface. It is obvious that if the attracted point traverses the 
surface obliquely instead of normally, viz. if the arc s cuts the surface obliquely, 
there is the like abrupt change in the value of 
Reverting to the original form of the two equations, and attending to the relation 
Q'-f-Q"=0, we obtain 
dW' dW" _ -4(Vj) s+1 
da' ds" r(^-i) & 
or, what is the same thing, 
r(^-j) /dW dw"\ 
Z— “4(ri) s+1 \ ds' + ds" ) 
(C) 
11. I recall the signification of the symbols: — ■V , ,V ,A are the potentials, it may bedifferent 
functions of the coordinates (a . . . c, e) of the attracted point, for positions of this point 
on the two sides of the surface (as to this more presently), and W', W" are what V', V" 
respectively become when the coordinates [a . . . c, e) are replaced by (x ... z, w ), the coor- 
f dW' dW" . 
dinates of a -bint N on the surface. The explanation of the symbols ~^r is given 
a little above ; § denotes the density at the point (x . . . z, w). 
12. The like remarks arise as with regard to the former distribution theorem (A) ; 
the functions V', V" cannot be assumed at pleasure ; non constat that there is any dis- 
tribution in space, and still less any distribution on the surface, which would give such 
values to the potential of a point (a . . . c, e) on the two sides of the surface respectively ; 
but assuming that the functions V', V" are such that they do arise from a distribution 
on the surface, or say that they satisfy all the conditions, whatever they are, required in 
