and this is in accordance with the general form- differential co-efficient of 1 or x-3 = 54. Tabulating these results, we have differential co-efficient of x-1 or 3 、。 and these come under the general form differential co-efficient of x-" or 1 xn 55. Let us now refer again to the function of the 1 1 form and take as an example of the function of X29 32 that form, where 3 is the variable; and let 3 receive small increments, as before, of 0001. Then, if we take the decimal out to a larger number of places, we shall find that the successive values of the function and the first and second differences become (see Art. 49) the second difference being positive, inasmuch as - 00000740554 is greater than -00000740631. Now the ratio of the rate of variation of the first difference to (the rate of variation of the variable)2 is the second differential co-efficient of the function; and the rate of variation of the first differences is given by the second difference. Therefore, we have, when 3 is the variable, 56. This result might have been obtained independently from the first differential co-efficient, for differential co-efficient of -3=-2x(-3)=2x3 This is of the general form— second differential co-efficient of 1_n(n+1) 34 xn+2 XII. Newton's Lemmas VI. and VII. 57. "If an arc ACB be subtended by the chord AB, and have the tangent A TD at A; then if the point B move up to 4, the angle BAD will diminish indefinitely and ultimately vanish." Draw the tangent BT at B; then the angle BTD continually diminishes as B approaches A, and ultimately vanishes. Therefore, a fortiori, the angle BAT which is less than BTD, continually diminishes and ultimately vanishes-i.e., the ultimate direction of the B A arc, chord, and tangent is the same, and is identical with that of the tangent A TD. 58. Definition. The subtense of an arc is a straight line drawn from one extremity of the arc to meet, at a finite angle, the tangent to the arc at its other extremity. 59. "If BD be a subtense of the arc ACB, and B move up to A, then will the ultimate ratio of the arc ACB, the chord AB, and the tangent AD be a ratio of equality." Let AD be produced to some fixed point d, and, as B moves up to A, suppose db always drawn through d, parallel to DB, to meet AB produced in b. Also on Ab suppose an arc Acb to be described, always similar to ACB, and having therefore ADd for its tangent. d Then, by similar figures, we shall always have and since this is always true, it is true in the limit, when B moves up to A. But, when B moves up to A, the angle bad vanishes, and therefore the point b concides with the point d, and the lines Ab, Ad, and therefore Acb, which lies between them, are equal. Hence also the arc ACB, the chord AB and the tangent AD, which are always in the same proportion as Acb, Ab, and Ad, are ultimately equal. Hence, in all reasonings, when the arc is very small indeed, the arc, the chord, and the tangent may be used indifferently for one another. XIII. Differential Co-efficient of the Trigonometrical Functions (Geometrically). 60. Let O be the centre of a circle, whose radius is 1, and in the arc of the quadrant AB take any point N B MM P, and join OP; and from P draw PM perpendicular to 40. Take any other point P' very near to P, on the arc, and draw P'M', PN perpendicular to AO and P'M. Then as P' moves up to P and ultimately coincides with it, the arc PP', the chord PP', and the tangent at P coincide; or, in the immediate neighbourhood of P, may be used indiscriminately, the one for another. Since the radius of the circle, viz. OP, is 1, it follows that Now, as the arc AP increases (i.e. as the angle POA increases) from AP to AP', it receives a small increment PP' and the sine of POM, viz. PM, receives a small increment P'N; and the ratio of the rate of variation of the sine (the function) to the rate of P'N variation of the arc (the variable) is ; and this is PP true for any position of P', and is therefore true when P' moves up to P; and then PP' becomes a tangent and the angle OPP'=90°. Therefore, as the angle POM (i.e. the arc AP) receives very small increments, the differential coP'N efficient of sin POM=: PP' =sin P'PN =cos NPO =cos POM. And this is of the general form differential co-efficient of sin x cos x. Again, the variation in the cosine of POM is represented in magnitude by OM-OM', or MM'; |