738 PEOEESSOE CAYLEY ON PEEPOTENTIALS. 
and we have as before, 
r (i* + g) //> 7iH /i or a?y 
e-(ri)*r( S +i) (/• • • 7i ) [}~f* ■ ■ ■ -*) • 
101. Hence in the formula 
V— f §dx. . .dz 
J {(a-x)*...+(c-zr+e *}** +9 
_ ie ^ 9 ’ ‘ 
§ has the value just found, or, what is the same thing, we have 
[ (-?■ 
z q \ q 
J | {a— a?) 2 . . 
. + (c-*) 2 + e 2 p s+2 
over ellipsoid • • • +|g=l, 
= (r r'(i,+g7 1) (/• • • 7 *)j 'j-'-V+f ■ ■ ■ t+hT'dt. 
102. We may in this result write e=0. There are two cases, according as the 
a 2 c 2 
attracted point is exterior or interior: if it is exterior, j? . . . +p> 1? $ will denote the 
positive root of the equation y 2 + g • • • +/^rg = l ; if it be interior,^ • • • +^2 < 1? $ will 
be=0 ; and we thus have 
1 f h* 
dx ... dz 
^-x)K.. + {c-zf\^ 
== ^f ~ (^ + g) 1 ^ (/• • • 7 0j] ^" ?-1 (^+/ 2 • • • t+tf^dt, for exterior point^ 2 . . . +^>1, 
= ~ r^ + g)^ (/* ‘ ^ 2_1 (*+/ 2 • • • t-\-h 2 )~Ht, for interior point Jg . . . +%< 1 ; 
but as regards the value for an interior point it is to be observed that unless g be nega- 
tive (between 0 and —1, since 1 + g is positive by hypothesis) the two sides of the 
equation will be each of them infinite. 
103. Third example. We assume here 
where 
V= dt I“T 
Je 
1 = 1 
T=r s " , (^+/ 2 . . . t+h 2 )-\ 
