PROFESSOR CAYLEY ON PREPOTENTIALS. 
683 
9. I consider positions of the point P on the two opposite sides of the point N, say 
at the normal distances a', a", these being positive distances measured in opposite direc- 
tions from the point N. The function V, which represents the potential of the surface 
in regard to the point P, is or may be a different function of the coordinates {a . . , c, e) 
of the point P, according as the point is situate on the one side or the other of the 
surface (as to this more presently). I represent it in the one case by V', and in the other 
case by Y" ; and in further explanation state that a' is measured into the space to which 
Y 1 refers, a" into that to which V" refers; and I say that the formulse belonging to the 
two positions of the point P are 
2(ru-) s+i 
dW 
da" 
=Q!'— 
2(ir)* +i 
where, instead of Y', Y", I have written W', W" to denote that the coordinates, as well 
of P' as of P", are taken to be the values (x...z,w) which belong to the point N. The 
symbols denote 
dW' dW , , dW ■ 
-d?=lu cosa'.-.+^-cos/, 
dW dW" ,, , dW" 
-W—^ cosx "-+-dF cos /■ 
where (cos a! . . . cos y') and (cos a" . . . cos y") are the cosine inclinations of the normal 
distances a', a" to the positive parts of the axes of (x . . . z) ; since these distances are 
measured in opposite directions, we have cos u"= — cos a 1 . . . cos y"= — cos y'. If we 
imagine a curve through N cutting the surface at right angles, or, what is the same 
thing, an element of the curve coinciding in direction with the normal element P'NP", 
and if s denote the distance of N from a fixed point of the curve, and for the point P ' s 
becomes while for the point P" it becomes s—W's, or, what is the same thing, if 
s increase in the direction of NP' and decrease in that of NP 7 , then if any function 0 
of the coordinates (x . . . z, w) of N be regarded as a function of s, we have 
d@_d@ d®_ d® 
ds da ' 5 ds da" ’ 
10. In particular, let 0 denote the potential of the remaining portion of the surface, 
that is, of the whole surface exclusive of the disk ; the curve last spoken of is a curve 
which does not pass through the material surface, viz. the portion to which 0 has 
reference, and there is no discontinuity in the value of 0 as we pass along this curve 
through the point N. We have Q'= value of ^ at the point P', and Q"= value of 
at the point P" ; and the two points P', P" coming to coincide together at the point 
MDCCCLXXV. 4 Y 
