306 
DETERMINATION OF THE ATTRACTION OF AN 
[604 
viz. this is the value of the radial thickness R'S' of the shell; or, since the same 
process applies to the point R", we have 
R'S' — R"S" = m dm 1 
\/B 2 -AG 
m 
or, calling this, as above, A dm, the value of A is = 
v B' 2 — AG 
8. The points P and Q are connected by the condition that, for every direction 
whatever of the chord R'R", we have 
PR' : PR" = QR' : QR", 
or, what is the same thing, that the line QP bisects the angle R'PR". Taking 
PR' = p, PR" = p", the condition is p : r = p" : r"; and taking (a, b, c) as the 
coordinates of the point P, we have 
p' 2 = (£ + t'cl — a) 2 + (rj + rfi — b) 2 + (£ + r'y — cf 
= a 2 + 2 r U + r 2 , 
if, for shortness, 
a 2 = (%- a) 2 + ( V ~ b) 2 + (f - c) 2 , (= QP 2 ), 
U= a(!;-a) + /3(y-b) + y(t-c); 
and similarly 
p" 2 = a 2 -2r"U+r" 2 . 
The required condition therefore is 
a 2 2 jj (t 2 2 TT 
/n H / u ,, u, 
fyv A ry* fp * fy* 
viz. this is 
+ 2 U 
(-, + 4) = 0, 
\r r ) 
2 /l_i 
a U.'2 r // 2 
so that, omitting a factor, it becomes 
<r ! (V4)+2P=°, 
\r r ) 
„ -25 
that is, 
S -+2V = 0, or V = 
which must be satisfied independently of the values of a, /3, 7. 
9. Writing, for greater convenience, ~ = —0, the equation is U = — OB, viz. 
Qt 
substituting for U, B their values, this gives a+^ = 0, &c., or say,