PROFESSOR CAYLEY ON PREPOTENTIALS. 
733 
exterior point, «>/*; the value is 
differing only from the preceding one in the inferior limit z 3 —f ' 2 in place of 0 of the 
integral. We have ^ — q positive, and also -^s + g' positive; viz. q may have any value 
diminishing from to — the extreme values not admissible. 
Annex IV. Examples of Theorem A. — Nos. 93 to 112. 
93. It is remarked in the text that in the examples which relate to the s-coordinal 
sphere and ellipsoid respectively, we have a quantity 0, a function of the coordinates 
(a. . . c, e) of the attracted point ; viz. in the case of the sphere, writing a 2 . . . -{-c 2 =z 2 , 
we have 
X 2 e 2 
/ 2 + < 
and in the case of the ellipsoid 
= 1 , 
c* 
'¥+6 
e* 
+ T = 1, 
/ 2 + 6 - 
the equation having in each case a positive root which is called 3. The properties of 
the equation are the same in each case ; but for the sphere, the equation being a quadric 
one, can be solved. The equation in fact is 
and the positive root is therefore 
6=\{e 2 -\-fc 2 —f 2 +\/ (e 2 -}-z 2 —f 2 ) 2 + ^e 2 f 2 [ • 
Suppose e to gradually diminish and become =0; for an exterior point, x>f, the 
value of the radical is =x 2 —f\ and we have for an interior point, ?-<f the 
_j_ K 2 
value of the radical, supposing e only indefinitely small, is =/' 3 — a 3 -f y 2 _ ^ e 2 , and we 
have 3 =-| e 2 ^ 1 4 y 2 ~ l ~ , =~J—^ or, what is the same thing, ^ = ^1— viz. the 
positive root of the equation continually diminishes with e, and becomes ultimately =0. 
If « or e be indefinitely large, then the radical may be taken = e 1 + *. 2 , and we have 
0 indefinitely large, =e 2 -\ -y 2 . 
94. Every thing is the same with the general equation 
/ 2 + < 
...+ 
A 2 -f 0 
= 1 
the left-hand side is =0 for 0—zn , and (as 3 decreases) continually increases, becoming 
infinite for 3 = 0; there is consequently a single positive value of 3 for which the value 
is =1 ; viz. the equation has a single positive root, and 3 is taken to denote this root. 
5 e 2 
