Let a be the length of AB, b that of BC, r that of AC; and at time t let them make angles o, x, 0 with some fixed line in the plane of motion. Then r cos e = a cos + b cos X, r sin = a sin &+b sin X. But & and x increase uniformly. Hence $=mt + a, x= nt + B, where m, n, a, ß, are constants. Thus r cos 0 = a cos (mt + a) + b cos (nt + B), r sina sin (mt + a) + b sin (nt + B). These are the general equations of Epicycloids and Hypocycloids; and from them all their properties may be derived. We confine ourselves to one or two very simple cases. (1) Let m=n, a = b. (This is the composition of two equal circular motions, in the same direction and of equal period.) We have This also denotes uniform circular motion, and of the same period, and in the same direction, as the components. (2) Let m=- n, a = b. (Here we compound equal circular motions, of equal period, but in opposite directions.) As before we have and this denotes vibratory motion in a definite straight line. 31. In any system of moving points, to determine the relative from the absolute motions; and vice versa. Let x1, Y1, 21, X2 Y2, z, be the co-ordinates of two of the points, x, y, z the relative co-ordinates of the second with regard to the first, u1, v1, w1, u, v, w, the velocities of each parallel to the axes, u, v, w the velocities of the second relatively to the first. Then 2 The second group may be derived from the first by differentiation with respect to t Now, when the actual motions of the two are given, all the subscribed quantities are known. Hence the above equations give the circumstances of the relative motion. Or if the actual motion of the first, and the relative motion about it of the second, be known, we have xyz, u v w x1 y11, uv, w1, to find the other six quantities for the actual motion of the second in space. A second differentiation proves the statement in § 26 regarding relative acceleration. 32. Some of the best illustrations of this part of our subject are to be found in what are called Curves of Pursuit. These questions arose from the consideration of the path taken by a dog, who in following his master always directs his course towards him. In order to simplify the question the rates of motion of both master and dog are supposed to continue constant; or at least to have a constant ratio. 33. As an instance of the curve of pursuit, suppose it be required to determine the path of a point P which continually with constant velocity u, moves towards another point Q which is describing a straight line with constant velocity v. The curve of course is plane. Take the line of motion of the second point Q as the axis of x, and let a denote its position at the instant when the co-ordinates of the first, P, are x, y. The axis of y is chosen as that tangent to the curve of pursuit which is perpendicular to the axis of x, and the distance between the points in that position is a. e, then by the conditions of the problem we have The mode of solution is precisely the same whether x or y be taken as independent variable: but y is to be preferred as it leads to less cumbrous expressions. Differentiating therefore with respect to y, we have But x = 0, y = a, together; which gives C = yo+1 a Hence 2(x+1)= a* (e + 1) + y2 (e − 1) e-1 ae e2-1' .(1), (2). This is the correct integral for all values of e except unity, when it ceases to have any meaning. To this case we will presently recur. There are two cases of curves represented by equation (2), 1st, e> 1, 2nd, e<1. In the first case Q moves the faster, and P can never over take it; the curve therefore never meets the axis of x, which indeed will be seen by (2) to be an asymptote. In the second case equation (2) becomes Hence the curve touches the axis at this point. The remainder of the curve satisfies an obvious modification of the question, whence it is called the Curve of Flight. {It is It is to be ( 1 + c ) = }) . ds ±y dy' or, by (1), where the sign is to be chosen so as to make the expression positive. When e> 1, this expression is infinite both for y= ∞ and for y=0. The minimum value is easily found to be When e<1, the distance vanishes, as we have seen it must, when y = 0. 34. When e = 1, the corrected integral of (1) is |