the Density of the Earth . 
I 
523 
APPENDIX. 
On the Attraction of the Mahogany Case on the Balls. 
The first thing is, to find the attraction of the rectangular 
plane c k fib (fig. 8 .) on the point a , placed in the line a c per- 
pendicular to this plane. 
Let a c = a, c k = b, c b = x, and let 
a 1 
and 
a b , 
a 1 -f x 1 ~ a 2 - 4 - x % 
v* f then the attraction of the line b fi on a, in the direction 
; and therefore, if c b flows, the fluxion of the at- 
’ a b x a (3 
traction of the plane on the point a, in the direction c b t 
b x 
x 
b ay 
— b njj 
O’ 
V a 1 -j-x 7, x ^a^+^+x 2, v / « z -j-.r a w ^ b 7 + - — ^b^w^-j-a 2 
V 1 v z> 
the variable part of the fluent of which = - 
and therefore the whole attraction = log. c -^~- x 
log. V -|- l+v\ 
so 
a c 
6 / 3 -f a (3 9 
that the attraction of the plane, in the direction c b , is found 
readily by logarithms, but I know no way of finding its attract 
tion in the direction ac, except by an infinite series. 
The two most convenient series I know, are the following : 
First series. Let 
7T, and let A == arc whose tang, is tt, 
B = A — 7T, C = B + j, D = C — ~ , &c. then the attrac- 
. .. ✓ — » a • 1 3Cto 4 , 
tion m the direction a c = v 1 — w x A-f- — rT? 7 “T TAftf 
&c. 
3X2 
