12 
ON THE PROPERTIES OF A CERTAIN SYMBOLICAL EXPRESSION. 
[2 
d? cl 
since, as applied to the function cf), + &c. is equivalent to 0; we have in 
this case 
da 2 rfò 2 
//... <f>(a-x, b-y ...)dxdy... 
2hh / ... 7r Jn 
r(*n) ' 
s; 
2-^ +1 .1.2 ... 79. in ... (^n + p) 
(V-^ 2 )^y a + ---r &•••); 
or the first side divided by hh t ... has the remarkable property of depending on the 
differences h 2 — h 2 , &c. only; this is the generalisation of a well-known property of the 
function V, in the theory of the attraction of a spheroid upon an external point. 
If in this equation we put cf) (a, b...) 
d 2 (f> 
(a 2 + 6 2 ...) 
Y n , which satisfies the required 
condition + &c. = 0, then transferring the factor a to the left-hand side of the sign 
da 2 
S, and putting in a preceding formula, a 2 = 0, /3 2 = h 2 - h 2 , &c. and rf + h 2 for y 2 , we 
obtain 
[[...(„ times) (»—)**- 
JJ {(a-x) 2 + (b-y) 2 
2 hh. 
_èn 
— f 
¿ ,n 1 dx 
VO7 3 + h 2 ) . T (|ll) J 0 V [W + h 2 + (h 2 - h 2 ) x 2 \ [y 2 + h 2 + (h t 2 - h 2 ) x*} ... (it - 1) factors] ’ 
where, as before, the integrations on the first side extend to all real values of x, y, &c., 
rjQ2 /£# 2 ^ 
satisfying ... < 1; t? 2 is determined by + &c. = 1; and a, b^.-h, h /} &c. are 
subject to + r- 2 + &c. • • • >1- 
A 2 h 2 
For n = 3, this becomes, 
(a — x) dx dy dz 
{(a - x) 2 + (b- y) 2 + (c — z) 2 
4irhh]n, ll a f 1 x 2 dx 
~ JW+fi] 0 V[{t + ^ + (V - ¿ 2 ) * 2 i W 2 + + (V - A 2 ) * 2 j] ' 
the integrations on the first side extending over the ellipsoid whose semiaxes are 
h, h /} h u , and the point whose coordinates are a, b, c, being exterior to this ellipsoid; 
, a 2 b 2 
also 
a* 0 J c- 
Tj 2 + k- + rf + h 2 y 2 + h r 
= 1 : a known theorem.