607] A MEMOIR ON PREPOTENTIALS. 
417 
6 now denoting either the positive root of the equation 
0+f> 
0 + h 2 6 + k 2 
= 0, 
or else 0, according as 
a- & e 2 
/» + - + & + s >1 or <h 
a- 
e- 
In the case ^ + ... + ^,<1, the inferior limit being then 0, this is, in fact, Jacobi's 
theorem (Crelle, t. xii. p. 69, 1834); but Jacobi does not consider the general case 
where l is not = 0, nor does he give explicitly the formula in the other case 
l-°’fi + - + h* + W> h 
150. Suppose k = 0, e being in the first instance not =0: then the former alter 
native holds good; and observing, in regard to the form which contains + w in the 
denominator, that we can now take account of the two values by simply multiplying 
by 2, we have 
dS 
f 
• f...k 
dx ...dz 
\{a — fxf + ... + (c - hzf + e 2 }*® ’ /... h J w {(« — x) 2 + ... + (c — z~f + e 2 p ’ 
(w on the right-hand side denoting 1—w 2 —and the limiting equation being 
j2+---+p =1 )> each 
= 2 Kf S?i i -tfp-■■■ - 
C e U- 1 
- — ^ = 0, which 
where 6 is here the positive root of the equation 1— $ + /¿,2 0 
is the formula referred to at the beginning of the present Annex. We may in the 
formula write e = 0, thus obtaining the theorem under two different forms for the cases 
qr2 
+ ... + ^- > 1 and < 1 respectively. 
Annex X. Methods of Lejeune-Dirichlet and Boole. Art. Nos. 151 to 162. 
151. The notion, that the density p is a discontinuous function vanishing for 
points outside the attracting mass, has been made use of in a different manner by 
Lejeune-Dirichlet (1839) and Boole (1857): viz. supposing that p has a given value 
f{x,.., z) within a given closed surface S and is = 0 outside the surface, these geometers 
in the expression of a potential or prepotential integral replace p by a definite integral 
which possesses the discontinuity in question, viz. it is =f(x,..,z) for points inside 
c. ix. 53