The difference between this and _xy_ is _xdy_ + _ydx_ + _dx.dy_. But _dx.dy_ is, as we have seen, a differential of the second order and may be neglected. Therefore _dxy_ = _xdy_ + _ydx_. In like manner _dx_² = (_x_ + _dx_)²-_x_² = 2_xdx_ + _dx_², which last term may be neglected, and _dx_² = 2_xdx_. In this way the differentials of all manner of functions and equations of symbols representing dimensions and motions may be found. Conversely the wholes may be considered as made up of an infinite number of these infinitely small parts, and found from them by summing up or integrating the differentials. Thus if we had the equation _xdy_ + _ydx_ = 2_zdz_ we know that the left-hand side is the differential of _xy_, and therefore that by integrating it we shall get _xy_; while the right side is the differential of _z_² which we shall get by integrating it. The relation expressed therefore is that _xy_ = _z_², or, in other words, that a rectangle whose sides are _x_ and _y_ exactly equals a square whose side is _z_. [Illustration: FIG. 1. FIG. 2. FIG. 3.] The use of this device in assisting calculation will be apparent if we take the case of an area bounded by a curved line. We cannot directly calculate this area, but we can easily tell that of a rectangle. Now it is evident that if we inscribe rectangles in this area ABC, the more rectangles we inscribe the less will be the error in taking their sum as equal to the curved area. This is apparent if we compare fig. 2 with fig. 3. Suppose we take a point P on the curve, call BN = _x_ and PN = _y_, and suppose N_n_ to be _dx_, the differentially small increment of _x_, and _pq_ = _dy_ the corresponding small increment of _y_. The area of the rectangle P_qn_N = PN × N_n_ = _ydx_, and differs from the true curvilinear area P_pn_N by less than the little rectangle of P_q_ × _pq_ or of _dx_._dy_. But, as we have seen, if we push our division to the first infinitesimal order, or make N_n_ and _pq_ differentials of _x_ and _y_, _dx_._dy_ may be neglected—i.e. multiply the number of rectangles indefinitely, and the sum of their areas will differ from the true area inclosed by the curve by an error which is evanescent. If then _x_ and _y_ are connected by some fixed law, as must be the case if the extremity of _y_ traces out some regular curve, the relation between them may be expressed by an equation, which will remain one however often it may be differentiated or again integrated, and whatever modifications or transformations it may receive by mathematical processes which do not alter the essential equality of the two sides connected by the symbol of equality =. Thus by differentiating and casting off as evanescent all differentials of a lower order than that which we are working with, we may arrive at forms of which we know the integrals, and by integrating get back to the results in ordinary numbers, which we were in search of but could not attain directly.