curves. areas; or, Since this is true of each strip, it is true of the whole [x'de + ['de = [(X'+dz X'dx+Edx E')dx * The two the limits of x being taken the same in all three curves. 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 +='+' + etc.)dx = (X'dx + [#dx + f· 'dx + etc. If the separate integrals on the right-hand side of the last equation be called X, E, X, etc.; then the differential view of the same proposition is that the 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. curve gives a definite value there is a definite value of corresponding value of λ, and the first of l; it follows that for each 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 definite. Thus A is a definite function of 7. are 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 8X are the projections of SL any very short length of the second curve lying partly on each side of the point where the tangent is drawn, then λ'= δλ 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 7-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 1, 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: Let L= 12 and λ= L3 .·.λ = 76. From SS 64 and 71 we know that L'=27 and X' = 3L2 = 3/4 Therefore, since 1/sin l=cosec 7, and 1/cos l = sec 1, 86. Reciprocal of a Function.-The second and the last of these illustrations are special cases of the semi-special semi-general case-a very important one Dividing by λ, that is, multiplying by L, we obtain this in the 87. Product of two Functions.-Retaining the notation of the last two articles, one particular function, to which we may apply the rule of § 84, is the product AL. Thus If X, E and x be used for the three mutually dependent variables, instead of the letters A, L and 1; and if X' and E' be the x-gradients of X and E; then the above is written This extremely useful result may be easily proved directly by the |