. An elementary treatise on the differential and integral calculus. erential of the first differential is the second differ-ential, represented by d2y, dhi, etc., and read, seconddifferential of y etc. The differential of the second dif-ferential is the third differential, represented by d3y, d?u,etc., and read, third differential of y, etc. In like man-ner, we have the fourth, fifth, etc., differentials. Differen-tials thus obtained are called successive differentials. Thus, let AB be a right linewhose equation is y = ax + b\.-. dy = adx. Now regard dx asconstant, i. e., let x be equicres-cen
. An elementary treatise on the differential and integral calculus. erential of the first differential is the second differ-ential, represented by d2y, dhi, etc., and read, seconddifferential of y etc. The differential of the second dif-ferential is the third differential, represented by d3y, d?u,etc., and read, third differential of y, etc. In like man-ner, we have the fourth, fifth, etc., differentials. Differen-tials thus obtained are called successive differentials. Thus, let AB be a right linewhose equation is y = ax + b\.-. dy = adx. Now regard dx asconstant, i. e., let x be equicres-cent;* and let MM. MM, andMM represent the successiveequal increments of x, or the dxs,and RP, RP, RF the corre-sponding increments of y, or the dys. We see from the figure that RP = RP = RP ;therefore the dys are all equal, and hence the differencebetween any two consecutive difs being 0, the differentialof dy, i. e., d2y = 0. Also, from the equation dy = adx wehave (Py = 0, since a and dx are both constants. Take the case of the parabola y2 = 2px (Fig. 7), frompdx. M M M NT Fig. 6 which we get dy = Regarding dx as a constant, we * When the variable increases by equal increments, i. e., when the differential isconstant, the variable is called an equicrescent variable. n EXAMPLES. have MM, MM, MM as the successive equal incrementsof x9 or the dxs; while we see fromFig. 7 that RP, RP, RP, or the dy% are no longer equal, butdiminish as we move towards theright, and hence the difference be-tween any two consecutive dys is anegative quantity (remembering thatthe difference is always found bytaking the first value from the Art. 12). Also, from the equa- tion dy = - dx we see that dy varies inversely as y. The student must be careful not to confound d2y withdy2 or d(y2): the first is second differential of?/; thesecond is the square of dy; the third is the differentialof y2, which equals 2ydy.
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