As explained below in § 83, the integral (X+X'x)dx= Xdx+ X'xdx. Therefore the result of this article may be written This is an important "Reduction Formula."* 81. x-Gradient of X/x and Inverse Integration. In fig. 18 a curve is drawn whose ordinates are called x and X, any function of x. From two points x,X and (x+dx), (X+dX) on this curve are drawn two straight lines to the origin 0; and on these two lines lie the upper extremities of verticals drawn at the horizontal distance 1 from 0. Evidently these last verticals measure the ratios and X + 8X The difference between them is the increase of the ratio due to the increase Ꮳ Sx of x, and is marked 8( in the figure. It is less than the small height aa in the ratio of 1 to x; and this height aa is less than 8X by bb. This small height bb bears the same ratio to Sx as (X+8X) bears to (x+dx). Thus * See Classified List, I. 8 and II. K. 82. Commutative Law.-If kX'8x is to be integrated, where to each X's the same constant multiplier k is to be applied, it is evidently allowable to sum up first the series of products X'sx, and then to multiply this sum by k. Symbolically written this is Jkx dx kX'dx=kX'dx taken between the same limits in either case. * Reverting to the graphic representation of integration in fig. 5, the proposition means that if there be two curves drawn, of which one has at each x its height k times the other, then the first has at each x its gradient also k times as steep as that of the other. 83. Distributive Law.-If there be two curves such as in fig. 6, the height of one being called X' and that of the other, then a third curve may be drawn, of which the height is (X'+E'). The area under the first curve is X'dx; that under the second is Edx; that under the third is (X'+E')dx. For each Sx at the same x, the area of the narrow strip (X' +E')dx for the third curve equals the sum of the two strips X'dx and 'dx for the first two * See Classified List, I. 4. curves. areas; or, Since this is true of each strip, it is true of the whole the limits of x being taken the same in all three curves. The two curves may represent entirely different functions of x, subject only to the one condition that they must be of the same kind, it being impossible to add together quantities of different kinds. In both integration and differentiation this proposition is more frequently used by way of splitting up a whole integral into parts easier to deal with taken separately than by the converse process of combining parts into one whole. It may be extended to the morë general formula If the separate integrals on the right-hand side of the last equation be called X, E, A, etc.; then the differential view of the same proposition is that the =x-grad.X+x-grad.E+x-grad.+ etc. Evidently the proposition of § 82 is only a special case of this in which X'E' = ', etc., etc. 84. Function of a Function.-In fig. 19 there is drawn a curve, the horizontal and vertical ordinates of which are called 7 and L. Thus L is a function of 1, the nature of the function being graphically described by this curve. A second curve is drawn, the vertical ordinates to which are the same L's as for the first curve (plotted and measured to the same scale), and whose horizontal ordinates are called A. A is a function of L, the curve graphically characterising the form of the function. A is a quantity which may possibly be of the same kind as 7, and, if so, it might be plotted to the same scale. But the general case is that in which A is not of the same kind as 7, and cannot possibly, therefore, be plotted to the same scale, although in the diagram it is measured in the same direction. Since for each given value of L the second curve gives a definite * See Classified List, I. 5. corresponding value of A, and the first curve gives a definite value of 7; it follows that for each value of 7 there is a definite value of A. In general there may be more than one value of λ for each 7; but all the values of A corresponding to one given value of 7 are definite. Thus λ is a definite function of 7. In the figure the l-gradient of L is represented graphically by a height obtained by drawing a tangent at the point /L, and plotting horizontally from this point a distance representing to the proper scale unity. This gradient is called L' in the figure. The L-gradient of A is similarly represented, the unit employed being measured vertically, and not being the same as that used in finding L', because the scales involved are different. It is marked X'. The two gradients shown in the figure are for the same value of L; that is, the tangents are drawn at points at the same level in the two curves. If SL and 8 are the two projections of any very short length of the first curve lying partly on each side of the point where the tangent is drawn, L'= If SL and SX are the projections of SL any very short length of the second curve lying partly on each δλ side of the point where the tangent is drawn, then X' = SL If the two short arcs on the two curves be taken so as to give the same vertical projection, that is, the same SL, as is shown in fig. 19 by the dotted lines; then in the product L'X' the SL cancels out. taken minutely small. In words this is :-The l-gradient of A equals the L-gradient of A multiplied by the l-gradient of L. dF(X) If we use the notation x, X, and F(X) instead of 7, L, and λ ; and if by F(X) we understand the X-gradient of F(X) or the same is written dX ; 85. Powers of the Variable; Powers of Sin and Cos.-This general proposition is one of the most fruitful of all laws in producing useful results when applied to particular functions, as will be seen in the next chapter. Simple illustrations of its meaning are the following :- • . λ = 76. |