and also that first differential co-efficient of x", where x is the variable, is nîn-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 × 2 × 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 0001; the through in Now let this space fallen Space. = 16.00320016 ft. 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. 49. Now take 1, and let it increase by small increments of 01, then in the first column of (1) will be found the reciprocals of the successive values of the variable 1; in the second column, the squares of these reciprocals; in the third column, the equivalents of these squares. It will be seen from the first and third columns that, as the variable 1 increases, the function (viz., the square of the reciprocal) decreases, therefore the differences (the fourth column), which are obtained from the numbers immediately above and below in the column to the left (the third), are negative, and that these differences are approximately in each case '02. Therefore the ratio of the rate of variation of the function to the rate of variation of the variable or, the differential co-efficients of the denominator is the variable: = 1 129 2 13. where the 1 in receives successive column, as before, 50. Now in (2) the number 2 increments of 001. The first represents the reciprocals of the successive values of the variable, the second column the squares of these reciprocals, etc.; and it will be seen that the first difference in each case is 00025 approximately; and the ratio of the rate of variation of the function to the rate of variation of the variable 51. Similarly from (3) the ratio of the rate of variation of the function to the rate of variation of the variable and, generally, it will be found that the differential 1 co-efficient of or x-2 is 52. Again Function. 1 2 2 Differences. 2.001 -'0002497 -'0002493 -'0002492 ='4990019 2.004 From this it will be seen that the function is the reciprocal of 2, as it receives successive increments of 001 and the difference in each case is 00025 approximately. Therefore the ratio of the rate of variation of the function to the rate of variation of the variable -'00025 ·001 - 25 = 4 1 22 = ; and similar results will and, generally, this is in accordance with the general form differential co-efficient of x-1 or 1 1 XC x2 53. Further, let us take a function of the form Therefore the ratio of the rate of variation of the function to the rate of variation of the variable 03696269. == -*037, D |