140 Transactions.— Miscellaneous, 
Equation (7) will be the best to take for a special consideration. The 
equation is 
t 
2r = 
{2 
This may be read,—The diameter is equal to the square of the tangent 
divided by the distance fallen through. Let us double the velocity and the 
body would traverse the tangent AP in half the time, and it would have to 
fall the distance PB also in half the time. The velocity would be repre-, 
sented by 2v. The equation would then stand 
op 208 
Here the two numeral factors, 2 and 4, cancel one another, and the 
tangent is unaltered. It will be seen, also, that to give the same quotient, 
2r, f must be increased four-fold; that is, the velocity being doubled, the 
acceleration had to be increased in the ratio of the square of the velocity. 
Other cases can be made up in the same way. In all of them the acceler- 
ation would have to increase as the square of the velocity. 
Mr. Proctor observes, in effect, of the radius vector sweeping over equal 
areas in equal times, as follows: ‘‘ Because a line, which is attached to a 
fixed point at one end and at the other to a body in motion, sweeps over 
equal areas in equal times, it does not therefore follow that the body is going 
in any orbit. For if the body moved in a straight line, the line joining the 
body to the fixed point would still move over equal areas in equal times. 
Let there be any fixed point A, and another B at some distance from it, 
and join AB. Let now a body be projected from B along a straight line 
BZ, at right angles to the line AB. It will move with uniform velocity in 
this direction, and of course will move over equal distances in equal times. 
Let it move from from B to C in one second. Mark off on the line AZ 
number of spaces CD, DH, EF, ete., each equal to BC. Join AC, AD, AE, 
AF, etc. The body passes through the points C, D, EH, F, ete., in successive 
seconds. The triangles constructed on these bases, BC, etc., are all equal to 
one another, because they have equal bases and are between the same parallels. 
The line joining the fixed point with the body moving in the straight line 
AZ, will therefore sweep over equal areas in equal times. Tt is not neces- 
sary, therefore, for the body to move in any orbit, because the line joining 
it to a fixed point passes over equal areas in equal times. 
Let a body be held at rest at the point A and let a uniform constant force 
act in the direction AZ. This line AZ may be conceived as being vertical to 
the surface of the earth, or any other planet, or the sun. Let the body at 
the point A be set free to the action of the accelerating force, and let it be 
drawn, or fall through, the points B, 0, D, at the end of the first, second, 
and third seconds respectively. The formula 
s=3 ft 
is perfectly general. If the accelerating force be equal to the force of gravity 
