291 
as are terminated by Planes, &c. 
The quantity of matter in the solid so generated being given, 
and the nature of one of the curves as pr', to find what must 
be the other curve pr, so that the action of the solid, on a point 
at its vertex p, may be the greatest possible. 
Put x for the absciss pm,y' and y for the ordinates mr', mr; 
then by Prop. 4, the action of the solid will be 
4 f* a rc (tang. = j^==|==) ; and its mass is 4 fyy'x. 
Buty is a given function of x y suppose f (x). The quantity 
Arc (tang. 
y f (*) 
x V ' x f (jc) 1 
+ Cyf (x) 
is therefore to have its fluxion, with respect toy, made = 0: 
and this gives, for the equation of the curve pr, 
^==r C = o, or a: 2 — C 2 (x z + / ) 2 (x 2 + y + f 
(a*+y*)*a*+y % + i (x) 1 1 \ • ^ \ ® 
(x)*) = 0 . 
Ex. 1 . Let f ( x ) = ax, or pr' be a straight line, the equa- 
tion of pr must be x 2 — C 2 (a: 2 -f-y ) 2 [ ( i + tf s ) x'-\-y % j == o. 
Ex. 2. Let pr' be a circle, or f (a:) 2 = vkx — x', k being the 
radius, then jc 2 — O (a: 2 +y ) 2 (zkx -J-y ) = o, is the equation 
of the ether curve. 
Ex. 3. If pr' is a parabola, or f (a:) 2 = ax, the equation of 
pr is x a — C z (x* -f-y )* (ax -}- x* + y) = °* 
Scholium . 
In Prop. 27, after having found the action of the solid there 
treated of, we derived, as corollaries, the action of parabolic 
and circular cylinders of infinite length, by separately making 
infinite the diameter of the circle and the parameter of the 
parabola. Perhaps it might therefore be supposed, that if we 
made k infinite in the second of the preceding examples, or a 
