MECHANICS, 27 
of the first second, the acceleration for the following seconds becomes 2G, 
3G ——-— iG, and the final velocity is V=7G. 
To determine the space, s, traversed by the body in the time, ¢, suppose é 
to be divided into infinitely small portions, and let the force operate only at 
the commencement of one of these divisions; if then the number of the 
Jt t 
divisions observed — 2, and the velocity at the end of the first division, - 
os Bee t 
= W, then the space traversed in the different divisions =W-. 
t t t 
DAME ip jer aW -, and s =W- (1+2+38---—+n). If nm be infinitely 
tn? t 
great, then s = we =n W rt and as nW must be the. final velocity, v, of 
ne) (GE 
the motion, s = z= 
following propositions respecting uniformly accelerated motion may be 
developed :—1, the final velocities attained at the expiration of different 
times are as these times; 2, the space described during uniformly accelerated 
motion, is half that which would be described if the motion had been equable 
and of the final velocity ; 3, the spaces traversed are as the squares of the 
times which have expired during the motion; 4, the spaces traversed in- 
successive equal times increase as the odd numbers, or as 1, 3, 5, 7, &c. 
The laws of uniformly varying motion may also be presented geometri- 
cally. Suppose the body to begin its motion from a state of rest at A 
(pl. 17, fig. 22); draw the straight line, AB, marking off upon it the equal 
parts, Aa, ab, be, and erecting the ordinates aa’, bb’, cc’, at the points of 
division. The abscissas, Aa, Ab, Ac, then represent the time elapsed since 
the beginning of the motion, and the corresponding ordinates, the final 
velocities. As these are all proportional to the aforesaid time, it follows 
that the line, AC, joining the ends of the ordinates, must be a straight line. 
Assuming the distances Aa, ab, bc, &c., as infinitely small, and drawing 
to AB the parallels a’b”, b’c”’, c'd’’, &c., small right-angled triangles result, 
whose sides, b'h’’, c'c’’, give the successive increase of velocity. The sur- 
face of the corresponding trapezoid has always an equal numerical value 
with the length of the path described by the accelerated motion; conse- 
quently the sum of all the trapezoids plus the small triangle, Aaa’, or the 
surface, Ahh’, represents the entire space traversed from the beginning. 
This triangle, however, is half the size of the rectangle which serves as the 
measure of the space traversed in equable motion, hence follows the propo- 
sition (No. 2) adduced above. 
The laws of the wnequably accelerated motion of bodies present many diffi- 
culties in their development. Suppose, in the first place, that it be desired, 
from the observed unequably traversed spaces and the corresponding times,to 
determine the velocity at the different points of the path described. To this 
end let AB (pl. 17, fig. 28) represent the axis of abscissas, AC the axis of 
ordinates of a system of rectangular co-ordinates, and A the starting point of 
201 
“gy and t= / om . From these investigations the 
