42. Now the second differential co-efficient might have been obtained from the first, for the differential co-efficient of 3x 12, if the 1 be supposed to vary, is 3×2×1, which is the same result as was previously obtained.

Similarly, the third differential co-efficient may be obtained from the second differential co-efficient, for the differential co-efficient of 6 × 1, if the 1 be supposed to vary, is 6.

43. Precisely similar results will be obtained from (2), but in this case the approximation will not be so far from error, inasmuch as the increments in the variable are not so small. For the first differential co-efficient we shall have to neglect thousandths, and for the second differential co-efficient hundreds-ofthousandths.

44. We shall obtain similar results whatever number we take as the variable: for instance, let us take 5, and let it vary by increments of '0001.

45. It may be noticed here that, as we found 2 to be the germ or essence of any system of variable squares,

so we find 6 to be the germ or essence of any system of variable cubes, and their successive differences, for

and, therefore, 00003, 0012, etc., are all of them functions of 6.

In the systems of squares, we found that the second differences received no increment-i.e., were constant; so, in the systems of cubes, we find that the third differences receive no increment-i.e., are constant.

46. If we adopt, for the fourth powers of any variable, a method similar to that already used for the squares and cubes, we shall arrive at analogous results; for instance, if 7 be supposed to receive small increments, then

first differential co-efficient of 74 = 4 × 73,

second
third

=4×3× 72,
=4×3×2×7,

and, regarding the fourth differential co-efficient as the

ratio of the rate of variation of the third difference to (the rate of variation of the variable),

fourth differential co-efficient as 7a=4×3×2

=24;

and 24 will be the essence or germ of any system of fourth powers.

47. Now, we have found that

the germ of second power=2=2×1='2,

and so it will be found generally, that the germ of the

nth

power

=n.n − 1...3.2.1= |»,

and also that first differential co-efficient of x", where x is the variable, is nx2-1.

NOTE.-Referring to Art. 30, it follows that the ratio of the rate of variation of 3 times the square to the rate of variation of the variable = 3 x 2 x side: 1; and of n times the square=n × 2 × side: 1; therefore the differential co-efficient of ax2=2ax, and the differential co-efficient of axnnaxn-1.

X. Method of Differences applied to the Motion of a Falling Body.

48. Let us apply this method of differences to the motion of a falling body.

In 1′′ a body falls through 16 feet. 1" receive increments of

Now let this

0001; the

space fallen

through in

From this we see that the ratio of the rate of variation of the function (the space fallen through) to the rate

of variation of the variable (the time)=

=

*0032

·0001

=32, omit

ting the figures in the seventh and eighth decimal places.

Now the first differences give the space fallen through in each successive interval of 0001", and the ratio will be more nearly correct the smaller we make the increments,

But these first differences are themselves receiving increments as the time increases, and the second differential co-efficient gives the ratio of their rate of variation to (the rate of variation of the time)2, viz. :

and this ratio has the same value, however small the increments be made.

Therefore, we may say that, at any instant, the space fallen through is increasing by some function of 32, and that that increase is, at that instant, also itself increasing by some function of 32-32 being the germ or essence of the system of spaces fallen through, and also of the differences.

XI. The Differential Co-efficients of an Inverse Function.