Dr. Young's remarks on the reduction 
9 ° 
centres of attraction will be to their distance b as fr to the 
oblique distance y/ ( 1 — 2 bgx') ; it will therefore be ex- 
btx 
pressed by 5 and the sine of the very small 
angular deviation of the joint force from the radius will be to 
the line measuring the disturbing force as this last sine to the 
radius, the difference of the third side of the triangle from 
the radius being inconsiderable ; consequently the deviation 
7 r 
will be every where expressed by = d. Now 
in order to find where this is greatest, we must make its 
qxdx 
fxibfadx 
fluxion vanish, and o = . , ,, , 
(1 -\-bb — 2b$x) | 2 (i+ftft — 2b$x)j l > 
gx (1 + b 2 — 2 bgx) = %bPx, 3% 2 x — Qbc'x -{- (1 + 6 2 ) 9 X — 36, 
fx + fx = 3, and px = V (3 + [4rf) — '~^r > but . 
making b=i-c, '-±t b becomes i± lrg + “ = 1 
and c being very small, gx will be s/ (4 -j- 
2 + 
cc 
~T 
cc 
26 
cc 
26 
1/ [1 — 1 + 
cc 
2ft 
— ; whence fx = y/ (1 — [1 — 
= TP) • or sim P 1 y and 
cc 
p~ 
•5-ri' 
C == i/ztx. 
B. The sine of the greatest deviation of the plumb line will 
amount to d= .38 5 a being the disturbing mass, and c 
its depth. 
Since cx = 1 
+ 
cc 
(1— -6) e 4~ 
2 bgx = 26 
cc r 1 cc 
2 
cc 
, and 1 -{-66 — 26 
aftc 
V^ft) 
, whence — 
y^ft .27 j C 3 
c 2 -{ — — = and a6fr becomes 
*9. 2 
, or simply 
aftc 2a*/b o a \/b 
^27 cc ^ ^ rr 
o3 8 5 _±_; also a = 2.6i8r 2 c/, and c = ^/ (.385 -j). If the 
density were doubled throughout the extent of a sphere 
touching the surface internally, the radius being c, we should 
