^7 6 Mr. Knight on the Attraction of such Solids 
Prop. 27. 
Let pr be a circle, pr' a parabola ; to find the attraction of the 
solid on a point p at its vertex. 
Let the radius of the circle be k, the parameter of the para- 
bola a : then we have h — s/zkx — X 1 ; h' — V zx, and 
. r- , ✓ aX V zkx~X Z \ 
A = 4jx arc (tang. = v^== ]• 
c V zk—x 
, if we put c = 
A/' 
V zk-^a. 
, and 
or A = 4/x arc (tang. = 
taking the fluent by parts 
* C zk — x\ r , , C ^ 2 k — x\ 
A = 4.2: arc (tang. = — I — 4/x arc (tang. = z= =~ ) 
the last term of which, if we put z* for x, becomes 
c^f zk — 2 
— 4 fz z arc (tang. == 
taking the fluxion of the arc =/- 
and this, by actually 
8kcz z z 
, or, 
|(I+C*) Z z + 2kc z } ^ 2k — Z z 
by restoring the value of c, = xV a . V 2k 4 - « . f* r ——? f===== 
or by division 
= 4 VT . VHk + Z .f { -~J=r — — } 
t V 2 k—z z ( z + a ) V zk — Z z } 
4 V 01 s/ 2k x arc (sine = ^=J — 42 arc (sine = 
zk-\- ce. z \ 
V zk V'z^-i-al 7 
so that we have at last 
A = 4-r arc (tang. 
v'" 
x - 2/ — - ] -}- 4 v/ # V 2k -a arc 
V 2 k-\-a. 
sine 
v' x \ , . V zk -\-cz V x \ , ✓ „» 
7SI ~ 4,2 arc ( sme = ~W * 77rJ + corr (/3)> 
As this expression vanishes when x = 0, if the fluent is to 
