708 
PEOFESSOE CAYLEY ON PEEPOTENTIALS. 
Suppose that the attracting body is a circular disk radius E, having the origin for its 
centre (viz. that bounded by the curve a?-\-y 2 = E 2 ); then writing oc=r cos 0, y=r sin 8, 
we have 
„ C grdrdQ 
V_ jp + e 2 )^ +2 ’ 
which, if § is a function of r only, is 
r % rdr . 
| (' r 2_|_ e 2ji«+ 2 > 
and in particular if %=r s ~ 2 , then the value is 
1 dr 
( r 2 + e 2)^+2’ 
the integral in regard to r being taken from r= 0 to r=E. It is assumed that s— 1 is 
not negative, viz. it is positive or (it may be) zero. 1 consider the integral 
p r *- 1 dr 
J o ( r 2 + e 2)i S+2 5 
which I call the r-integral, more particularly in the case where e is small in comparison 
with E. It is to be observed that e not being = 0, and E being finite, the integral con- 
tains no infinite element, and is therefore finite, whether q is positive, negative, or zero. 
54. Writing r=e\/ v, the integral is 
the limits being 
E 2 
v^ s ~ l dv 
.V)i s+ i’ 
and 0. 
In the case where q is positive this is 
viz. the first term of this is 
vis-'dv 
(T+wji*+2 ; 
2 r &+ q y 
and the second term is a term expansible in a series containing the powers 2q, 2q-\-2, 
e 2 i 
&c. of the small quantity as appears by effecting therein the substitution v=~; viz. 
the value of the entire integral is by this means found to be 
)vas + q) 1 
ii x q ~ l dx ) 
\T(±s+q) J 0 {l+x)i s+q y 
55. In the case where q is =0, or negative, the formula fails by reason that the ele 
becomes infinite for indefinitely large values of v. 
v ^ s — * d/V 
ment t v , of the integrals 
(l+v)i s+ 2 
rs 
Eecurring to the original form \ ~ it is to be observed that the integral has a 
Jo v + e ) s 2 
