depends by known relations on some arbitrary independent variable;—for example, in a given curve MN, fig. 15, it is required to determine the point in which the ordinate p m is the greatest possible. In this case, the curve, or function expressing the curve, remains the same; but in the other case, the form of the function whose maximum or minimum is required, is variable;

for, let M, N, fig. 16, be any two given points in space, and suppose it were required, among the infinite number of curves that can be drawn between these two points, to determine that whose length is a minimum. If ds be the element of the curve, ∫ds is the curve itself; now as the required curve must be a minimum, the variation of ∫ds when made equal to zero, will give that curve, for when quantities are at their maxima or minima, their increments are zero. Thus the form of the function ∫ds varies so as to fulfil the conditions of the problem, that is to say, in place of retaining its general form, it takes the form of that particular curve, subject to the conditions required.

58. It is evident from the nature of variations, that the variation of a quantity is independent of its differential, so that we may take the differential of a variation as d.δy, or the variation of a differential as δ.dy, and that d.δy=δ.dy.

59. From what has been said, it appears that virtual velocities are real variations; for if a body be moving on a curve, the virtual velocity may be assumed either to be on the curve or not on the curve; it is consequently independent of the law by which the co-ordinates of the curve vary, unless when we choose to subject it to that law.