An introduction to the summation of differences of a function; an elementary exposition of the nature of the algebraic processes replaced by the abbreviations of the infinitesimal calculus, by B. F. Groat.

AREAS AND LIMITS OF SUMS

35

28. It was shown in the preceding part of this chapter how
the solution of the problem of areas can be made to depend
upon the solution of the problem of a limit of a sumn. Many of
the most important problems in mieclianics can be made to
depend upon a limit of a sum. We will take a simple example.
Suppose we wish to find the space described in a given time by
a freely falling body. We know by experiment that the velocity
gains a definite number of units in each unit of time. This
gain per second in velocity is called the acceleration of the body,
and is represented by g. Near the surface of the earth g= 32.2,
in the foot-pound-second system of units; this meaning that the
velocity increases 32.2 feet per second in a second.
SOLUTION. Let v0 be the velocity at the beginning of the given timeinterval, and measure time from that instant, representing it by 0.
Divide the total interval, t, into nt equal parts, each __ A, and
represent the elapsed time at the beginning of the rth partial interval by,. = r — i AO. Then the velocities at the beginning and end of the rth
-interval are zvo + gor,, zo + gO,.++1
r=nV r=)+1
Whence (z0 + g,.) A0 < s < (zo +,,g0,) A; where s is the required actual space passed over during the time t.
n+1 n
But X ( 'o + g,.) A - - (7o + g,.) AO
r=2 r=l
= g(O,,+ - O) aO = gtAO
vanishes when AO = o, from which it follows that
n n n
S = -o E + 0) AO = A vL o 0 0O +O j0 E OA 0.
Now=l =1 v -- i
Now d o A 0' o a. or7O 0 AO = z,/,.?.=I