and shorter; and the limit towards which converges, as As At and ▲t are indefinitely diminished, and which is denoted by is the exact velocity at the instant of passing P. In the language of the differential calculus, the space is a function of the time and the velocity is the differential coefficient of the space with respect ds d t to the time, thus s= Φ t and =p't=v. It will be seen hereafter that, the velocity (v) itself is a function of the time (t). This is the process called "differentiation." Should the velocity at each instant of time be known, then the distance 81 - 80, described during an interval of time t1 - to, is found by integration (see Article 29), as follows: 72. Components of Varied Motion.-All the propositions of the two preceding sections, respecting the composition and resolution of motions, are applicable to the velocities of varied motions at a given instant, each such velocity being represented by a line, such as PT, in the direction of the tangent to the path of the point which moves with that velocity, at the instant in question. For example, if the axes O X, O Y, O Z, are at right angles to each other, and if the tangent PT makes with their directions respectively the angles a, 3, then the three rectangular components of the velocity of the point parallel to those three axes are v cos a; v cos ß; v cos y. Let x, y, z, be the co-ordinates of any point, such as P, in the path AP B, as referred to the three given axes. If a point p be assumed indefinitely near to the point P, its co-ordinates will be x+dx, y+dy, z+dz, and if ds have the already assumed value, dx, d y, dz, will be its projections on the three axes; that is, the lengths bounded by perpendiculars let fall from the extremities of ds on the three respective axes. Then it is well known that UNIFORMLY VARIED VELOCITY. now by the Geometry of three dimensions cos2 a + cos2 ß + cos2 = 1. and hence these are related to their resultant by the equation 41 73. Uniformly-Varied Velocity.-Let the velocity of a point either increase or diminish at an uniform rate; so that if t represents the time elapsed from a fixed instant when the velocity was vo, the velocity at the end of that time shall be a being a constant quantity, which is the rate of variation of the velocity, and is called acceleration when positive, and retardation when negative. Then the mean velocity during the time t is If there be no initial velocity, that is, if the body start from a a t2 state of rest, then v = a t and s=· and these equations are illus " 2 trations of the use of the differential calculus; for first differentiate s with respect to t in the equation s= and there is obtained a t2 2' ds d t at, which is the first equation, then differentiate v=at, and there is obtained dv point, whose velocity is uniformly varied, at a given instant, and the rate of variation of that velocity, let the distances, A81, 82, described in two equal intervals of time, each equal to At, before and after the instant in question, be observed. Then the velocity at the instant between those intervals is where the variation of velocity = = A82-A81 and the rate of varia tion being either acceleration or retardation, as the velocity of the point is being increased or diminished, is that quantity divided by At. C F D 74. Graphical Representation of Motions. Since in uniform motion the space is equal to the product of the velocity and time, and since in geometry a rectangular area is the product of a base line and perpendicular, an uniform motion may be represented by a rectangular area, as in fig. 25, where A B represents a certain number of units of time, and A C a certain number of units of velocity per unit of time. It will be noticed that in uniform motion, the velocity or number of units of velocity at each unit of time is the same, as at A, B, E. Varied motion and uniB formly varied motion may also be graphically represented: in the first, the line CD will be a curve; and in the second, the line CD will form a constant angle with A B; hence in varied motion any ordinate, E F, depends upon the abscissa A E, and the mean velocity is the mean ordinate of a figure so formed, A E Fig. 25. T A E Fig. 26. or is the quotient of the area (space) divided by the base (time), whereas in uniformly-varied motion, the space described depends upon the initial and final velocities alone, and not upon the B intermediate velocities. Fig. 26 represents varied motion where the velocity at each point is re presented by the ordinate at that point, and the mean velocity is C equal to the area of the figure divided by the base A B. Fig. 27 represents Duniformly-varied motion, and it is evident that, in order to estimate the area of the figure ABCD, that is, the space, it is only necessary to consider the initial and final velocities. In these figures, if the velocity be null at any point, there will be no ordinate at that point: if the direction of motion change, this will be represented by a change of sign of the ordinate or velocity. A Fig. 27. B There is another method of graphically representing the motion of a point: in this the abscissæ represent the time, and the ordinates ACCELERATED MOTION. 43 at each point the space passed over in the corresponding number of units of time, or the distance of the point from a certain datum point. In this case the space described in any number of units of time is equal to the difference of the lengths of the ordinates at the corresponding intervals, and the velocity is proportional to the quotient of the difference of the ordinates divided by the difference of the abscissæ. 75. Varied Rate of Variation of Velocity.-When the velocity of a point is neither constant nor uniformly-varied, its rate of variation may still be found by applying to the velocity the same operation of differentiation, which, in Article 73, was applied to the distance described in order to find the velocity. The result of this operation is expressed by the symbols, and is the limit to which the quantity obtained by means of the formula 5 of Article 73 continually approximates, as the interval denoted by At is indefinitely diminished. In the fraction ds is the limit of the difference of either of the spaces As in equation (5), Article 73, and d·d s, is the limit of the difference of that difference, viz., As-As; that is, d in this fraction is represented by the minus sign (-) in the other, and ds by the limit of either of the quantities As, As2. Here in the language of the differential calculus, the velocity (v) is a function of the time (t), and the acceleration (a) is the differential coefficient of the velocity with respect to the time, thus vt and a = 't, or = d v dť Also the velocity, v, being the differential coefficient of the space with respect to the time, see Article 71; the acceleration a is the 2nd differential coefficient of the space with respect to the time, or v being 4't, a = 4" t. 76. Combination of Uniform and Uniformly Accelerated Motion. -Assume a pair of rectangular axes of co-ordinates. Let the uniform motion be represented by abscissæ along O X, and the uniformly accelerated motion by ordinates parallel to OY; let OB (=x)=vt, represent the space described in the time t with at 2 the velocity v, and let OC (=y): 2 , represent the space de scribed with a uniform rate of acceleration, a, in the same time t, see Article 73, then x2=v2t2 and Fig. 28. square of any abscissa bears a constant ratio to the corresponding ordinate, and the path of the point is known by Conic Sections to be a Parabola. The same follows for any axes of co-ordinates; but if the direction of the uniformly accelerated · motion be that of the uniform motion or directly opposed to it, the resultant direction will be the same as that of either motion, or will be that of the greater component. 77. Uniform Deviation is the change of motion of a point which moves with uniform velocity in a circular path. The rate at which uniform deviation takes place is determined in the following 1 A2 Fig. 29. 2 manner:- Let C, fig. 29, be the centre of the circular path described by a point A with an uniform velocity v, and let the radius C A be denoted by r. At the beginning and end of an interval of time ▲t, let A1 and A2 be the positions of the moving point. Then 2 The velocities at A, and A, are represented by the equal lines A1 V1=A2 V2=v, touching the circle at A1 and A, respectively. From A, draw A, v equal and parallel to A1 V1, and join Vv. Then the velocity A, V2 may be considered as compounded of A2v and v V2; so that v V2 is the deviation of the motion during the interval ▲t; and because the isosceles triangles A, v V2, CA, A2, are similar: 2 2 2 deduced by substituting the value of A, A, already found; and the approximate rate of that deviation being the deviation divided by the interval of time in which it occurs, is |