as are terminated by Planes , &c. 29J 
the equation of the curve, bounding the plane of greatest attrac- 
tion y is 
m-f-i 
a m x = ( x- -J- y* )~P > 
which is exactly the same result as that obtained by Mr. Play- 
fair, p. 203, on the supposition of homogeneity ; and this was 
to be expected ; for, though a certain condition of the density 
of the cylinder entered into the foregoing problem, yet when 
d vanishes, and the solid becomes a plane, we must evidently 
obtain the same result as if it had been arrived at by suppos- 
ing the cylinder homogeneous ; which in fact it will be when 
the length is evanescent. 
Nor is this observation to be confined to that particular case 
when the density is as / : if we had solved the problem on the 
supposition of any function of x, T, and t, for the density, it is 
easy to see that though different functions will give different 
results when d is finite, yet when the solid becomes a plane, 
and d—o y the equation will always be reduced to 
a m x = (x* -j-y*) 
Hence we may conclude, that, the solid of revolution which 
shall exercise the greatest attraction on a point in its axis, when 
the force is inversely as the mth power of the distance, and the den- 
sity either uniform, or any function whatever of x and T (T being 
the perpendicular let fall from any particle to the axis of the solid 
and x the distance between the foot of that perpendicular and the 
attracted point ) will have, for the equation of its generating curve , 
ro-J- 1 
a m x — (x 9 + y 9 ) 
Cor. 2. Nothing can be learned from the equation 
— + Cd 2 =r 0, 
( **+>*)— 
2 2 
