A MEMOIR ON PREPOTENTIALS. 
327 
607] 
where (cos a.', , cos 7') and (cos a.", .., cos 7") are the cosine-inclinations of the normal 
distances s', s' to the positive parts of the axes of (x, z); since these distances are 
measured in opposite directions, we have cos a" = — cos a!, .., cos 7" = — cos 7'. 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' if s become s + B's, while for the point P" it becomes s — B"s, or, what is the same 
thing, if s increase in the direction of NP' and decrease in that of NP", then if any 
function © of the coordinates (as, .., 2, w) of N be regarded as a function of s, we 
have 
d© rf© cZ© ri© 
ds ds' ’ ds ds" ‘ 
10. In particular, let © 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 
© has reference: and there is no discontinuity in the value of © 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 N, we have then 
Q' = 
d© 
dJ ’ 
d© 
ds ’ 
We have in like manner 
above may be written 
d© 
d© 
Q" 
“ ds" ’ 
ds 
dW' 
dW' 
dW" 
dW 
ds' ~ 
~ ds ’ 
ds" 
ds 
dW' 
d© 
2(r|) s+1 
ds 
ds 
r<*« 
-iV’ 
dW" 
d© 
2 (r4) s+1 
ds 
ds + 
-i) p ’ 
and the equation obtained 
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 
dW dV . . 
the value of , or say of , the first derived function of the potential m 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 
dV 
obliquely, there is the like abrupt change in the value of -j-.