308 Mr. Knight on the Attraction of such Solids 
A =4 (x + pT-t) arc (tang. = A v// 3 x+ y x‘) - + <p 
where <j> ■ — L ~p£] + + 1 ) > or = — 4 — — — 
v' 7 (i+ya 1 ) v'jS / @ ✓ — y(l+7a l ) 
arc (sine = ~y=~) as * or 7 is positive or negative. 
This is the action of an infinitely long cylinder on a point at the 
vertex of its transverse section, the equation of the said section 
being y 2 == «• ( ( 3 x -{- yP ) . 
Ex. If the base, or transverse section, is an ellipsis, or if 
b 1 b z 
y* = — ( ax—x 2 ), we have ot — — , /3 = a, 7 = — 1 ; and 
A = 4 (* + ^ji] a f c (‘ang. = A v^tx - x‘) + ^ 
4 a^b 
arc 
(sine = ^ 7r. When x ~ a, this expression i 
reduced to 
is 
a-i-b 
Scholium to Cor. 2, Prop. 30, page 281. 
If we would have a general expression for the attraction of 
such solids as the one we considered in the proposition, when 
the guiding curve is any conic section, or when 
y 1 =. cd ( fix -f- yx‘ ) , there arises at first (from the formula for 
the action of a rhombus ) 
A = 4 fx arc (tang. = Vr-|- ( 1 -f r * ) (/to -j- yx ') ) -f 
4 f* arc ( tan g- = h + ( 1 + d 2 ) a 2 ((lx + yx°)) — 27 rx, 
and by actually taking the fluent 
A = 4 (x + r |^) [arc (tang. = A v^tx + 7 F) 4. arc 
(tang. = £ '//x + /F) } - + 2X) *■ + <f + $>', 
