ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 175 
infinitely great; it is, however, unnecessary for our object to 
dwell on such exceptional cases, which can only occur in singu- 
lar points or lines, and thus not in parts of the surface, but only 
at the limits of parts. 
Lastly, it is self-evident, that it is in all respects with the 
quantity Z, or the integral ie be B a 
, as with Y; that is to say, 
if in the first coordinate axis the point O approaches infinitely 
near to the point P, this integral has the same limiting value Z°, 
whether the approximation be on the positive or on the negative 
side, and this limiting value is at the same time the value of 
S/S he “¢ ab for x = 0, if we integrate first with respect to c. 
18. 
Now, if we remember that the quantities -~ 7 o 
in all points of space not situated in the surface itself are un- 
conditionally the same as X, Y, Z, and that V varies everywhere 
continuously, it is easy to deduce from the results obtained in 
the last article, that at an infinitely small distance from P, or for 
infinitely small values of x,y,z, the value of V is expressed, 
within infinitesimals of a higher order, by 
V0 +a (X°—-27k)+y VY + 270 
if z is positive, or by 
V° + @ (X°4 20k) +y ¥°+ 27 
if # is negative, where V° denotes the value of V at the point 
P itself, or for = 0, y=0, z=0. Thus, if we consider the 
values of V in a straight line drawn through P, making with 
the three coordinate axes the angles A, B, C, and denote by ¢ 
an indefinite portion of this line, and by 7° the value of ¢ at the 
point P; then if ¢ — 7° be infinitely small, we shall have, within 
an eetesiscal of a higher order, 
V=V°+(¢—?°) (X° cos A+ Y° cos B+ Z° cos C F2 7 k° cos A), 
the lower sign holding good for positive, and the upper one for 
negative values of (¢ — ¢°) cos A; or = 
different values when A is acute, viz.— 
X° cos A+ Y° cos B+ Z cos C—2 7} cos A, and 
X° cos A + Y° cos B+ Z° cos C +2 7 f° cos A, 
according as dt is considered positive or negative. For the case 
has at the point P two 
