ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 171 
call these three values of X, X°, X', X", and the corresponding 
values of Q, Q°, Q’, Q”. 
Asr = ¥V ((a — 2)? + p*), we obtain, considering # as con- 
stant, 
h(a—2x) h(a—xz)pdp dh a—v da hp , | 
ari a Pidlo = i = Bs 
and consequently Q = 
dh a—@2 da he i! (a'— 2) ! 
dp. 2 -dp+ dp. Pi -dg ——~—_——~ + const. : 
where both the integrations are to be extended from p = 0 to 
p = ’, and the values of h, a, r for p = p! are denoted by fh’, a’, 7’. 
We must assume as a constant the value of — for p=0, 
which, if we designate the density at P by k°, becomes = — (° 
for a positive 2, and = +- k° for a negative x; as it is manifest 
that for p= 0 we have a=0,P =0,h=h°,x= +7. For the 
case of = 0, on the other hand, we must assume as constant 
the limitary value of =, as p decreases indefinitely ; this is=0, 
because a is an infinitesimal of a higher order than r. 
d Ae a—2x 
dp” 
an infinitesimal, whether we make 2 = 0, or infinitely small, = 
+«. If we decompose that integral ae 
odh a— 2 
Leaps aaa det Si a, Su 18 
: it is clear that what has been said holds res for the first mem- 
The value of the integral - de only differs by 
ber, if @ is infinitely small; and for the second, if a is infinitely 
great; and so for the whole, if 8 is an infinitesimal of a lower 
order than <. 
A similar conclusion holds good also in relation to the integral 
d 
ee. i = de, if the points of the surface which correspond 
to the determinate values of § form a curve having a finite cur- 
vature at P, so that aa may receive in the space here contem- 
plated a finite continuously varying value. I[f we put A for 
this value, then 
da _ WBS a 
ee : =2Ae+ ap Pats 
N 2 
