2S4 Mr. Knight on the Attraction of such Solids 
ing the subject connects it intimately with the preceding parts 
of this paper ; otherwise I should not have given the following 
problems a place here. 
Prop. 31. 
Suppose that a given quantity of matter is to be formed into 
a right cylinder of the length <id ; what must be the figure of 
its base, so that it shall attract, with the greatest force pos- 
sible, a point in its surface, and in the middle with respect to 
its two ends ? 
Let fig. 19 represent a section of the cylinder, at the at- 
tracted point p, parallel to its base. It is plain enough, that, 
whatever is the nature of the curve pab, we may draw a line 
pb from p, which shall divide the area into two equal and 
similar portions pabp, pcbp. 
Put the absciss pd = x, the ordinate ad = y: the mass of 
the cylinder is 4 dfyx\ and, by Prop. 4, its attraction on p is 
4/* arc (tang. = _^==). 
Let C be a constant quantity, and we have only to make 
the fluxion of the following expression, with respect to y, 
equal to nothing,* viz. 
Arc (tang. = 7^==) + Cdy: 
this gives 
C = 0, or x 2 — C 2 )x'-\-y*p (d 2 -\-x*~ fy*) = o. 
for the equation of the curve pab. Make y — o, and let a be 
the corresponding value of x ; the equation becomes 
1 .*=_ C V (^*+ a 2 ) =0, whence O = ^ fJ+Vy by substituting 
? Euleri “ Methodus, &c.” p. 42 and 185-6-7. 
