28 PHYSICS. 
motion ; the times may be taken as abscissas, the spaces traversed us ordinates. 
Assuming the parts Aa, ab, and c, of the axis of abscissas as infinitely smal, 
then the line Ad connecting the extremities of the ordinates, a’, b’, cannot be a 
straight line, but must be curved; the small triangles, a’/b’b", b'c'c”’, must alsv 
be unequal; consequently, the velocity of motion must change at ever; 
moment. Suppose, furthermore, that at any instant of motion, correspond- 
ing to the point c’, this motion suddenly becomes equable, then this new 
motion will be represented by a straight line, c’E, the prolongation of the 
chord of c'd’. As, moreover, the moving point in the instant when the 
motion is considered, during the elementary time c'd’ or cd, would have 
described the space d’d’’, it will by reason of the ensuing equable motion 
describe a space in the unit of time, determined by obtaining the ordinate 
mn tor c/m and c'n; the space mn then serves as the measure of velocity 
for uniform motion, and is thus the final velocity desired. 
c. Freely Falling Bodies and Projectiles. 
The theory of freely falling bodies is a consequence of the preceding pro- 
positions respecting uniformly accelerated motion. The force of gravity 
which here comes into account, must, if the motion be uniformly accelerated, 
be a constant force. It is known, indeed, that the intensity of this force 
diminishes as the square of the distance from the centre of the earth; as, 
however, the greatest space which can be traversed by a body is extremely 
minute, compared with the earth’s radius, it will involve no serious error to 
consider the action of gravitation within these limits as a constant force. 
The weight of the body is not taken into account in determining the laws 
of free falling, as gravitation acts uniformly upon all the atoms of a body, 
and although practically, weight does seem to be of account, the reason of 
this lies in the resistance of the atmosphere: all bodies fall with equal 
velocity in a vacuum. 
In the free falling of bodies, the two propositions may be brought into 
application—-that the velocities of a freely falling body are constantly pro- 
portional to the time expired, and that the spaces are as the squares of the 
times. It becomes necessary to determine the acceleration produced by 
gravitation, that is, the value of the space fallen through at the end of the 
first second, which can only be done by direct experiment. From carefully 
conducted experiments, it has been found that at a mean geographical lati- 
tude, and a height not too great above the level of the sea, the acceleration 
amounts to 9.81 metres (31 feet, 11 inches, 11 lines, English; 30/ 2” 7”, 
French; 31/ 3” 2’”, Rhenish). Calling this acceleration g, the body in the 
Cc 
in the second, g7® in the third, 3° 2° and the 
oO 
5 
first second traverses 1 5 ; 33 
9 bf 
/ . a 
entire space, s, fallen through in ¢ seconds is ¢ 9° 
Atwood’s machine is best adapted to demonstrate the correctness of re- 
sults obtained by these investigations. The entire instrument is figured in 
pl. 16, fig. 17; fig. 18 represents its upper portion on a larger scale. 
202 
