MECHANICS 683 Fig. 7. its focus S and its directrix MN. We suppose it to be placed with its axis vertical, and vertex upwards. Take any point P, join PS, M ^ and draw PM perpeu- " dicular to the directrix. Then (a) If PQ bisect the angle SPM, it is the tangent to the parabola at P. Let Q be any point in the tangent, and let QR, drawn parallel to MP, meet the curve in R. Then we have (b) PQ 2 = 4SP.QR. tli of 41. Now suppose a point, originally moving along PQ um-e- with uniform speed V, to have its motion accelerated in a ted direction parallel to MP, the acceleration being a, a con- tile stant. Then, after t seconds it would have moved along PQ through a space Vt, and parallel to MP through a space -i-aZ -. Hence, if R be its position at that time, PQ = V* , QR = ia< 2 . From these equations we find at once 9V 2 PQS.1JLQR. a This relation is of the same form as that already written for a parabola, and (as it does not involve t) it holds for every point of the path. Hence the point moves in a parabola whose axis is vertical, which touches PQ (the direction of projection) in P, and in which SP = V 2 /2a. But these three data determine the parabola. For we have only to draw PM vertical, make the angle QPS = QPM, and measure off the lengths PM and PS each equal to V 2 /2a. M is a point in the (horizontal) directrix, and S is the focus. Hence the path is completely determined. It is well to notice that, as V 2 = 2aPM, M is the point which the projectile would just reach if it were projected vertically upwards ( 29). amples 42. If the speed of projection be kept constant, while t h e direction of PQ alters in a vertical plane, S describes a circle about P as centre. This consideration enables us easily to find the direction of projection that a given object may be struck. Let (fig. 8) be the object. Join PO, and let it cut in B the circle MBS (whose centre is P). Draw ON perpendicular to the com mon directrix, and with radius ON describe a circle about O. This will (in general) cut MBS in two points F and F . These are the foci of the two paths by either of which the Fig. 8. projectile can reach 0. For by construction FO = ON, so that lies on the path whose focus is F. Similarly for F . To find the most distant point along PO which can be reached, with the given speed of projection from P, we have merely to note that, as O is taken farther and farther from P, F and F approach B, and finally coincide with it. If O be then at A, we have AT = AB, where AT is per pendicular to the directrix. Hence, if we produce AT to t so that T^ = BP, we have Ai! = AP. Draw through -t a line tm parallel to TM. Then A lies on the parabola whose focus is P and directrix mt. This parabola is the envelop of all the possible paths from P. Any point within it can be reached by two different paths. These "become coincident when the point lies on the curve; and no point outside it can be reached. motion pro- tile. Fig. 9. 43. Many of the most important cases of motion of a Accelera- point involve acceleration whose direction is always towards tion to a definite " centre " as it is called. In such cases the motion ! is obviously confined to the plane which, at any instant, e contains the centre and the line of motion of the point. Also the "moment" of the point s velocity about the centre remains constant. Here a slight digression is necessary. DBF. Given a directed quantity (a velocity, force, d c.) in Moment, a line AB (fig. 9). If a perpendicular OP be dratvn to A If from any point 0, the " moment " of the directed quantity about is the product of its amount by the length of the perpendicular. If the directed quantity be reversed, the sign of the moment is changed. The moment is, in fact, properly a directed quantity (or vector) perpen dicular to the plane OAB. And its numerical magnitude is double the area of the triangle OAB. [ 44. The convention usually made as to the sign-of rota- Conven tion about an axis is to regard it as positive when it is in tion as to the same sense as that in which the earth turns about Slgn of its axis, as seen by a spectator above the north pole. a"g u e j ar This is in the opposite direction to that of the hands of a velocity, watch. Hence the plane angle AOP (fig. 10), represent- &c. ing the change of direction of a line originally coincident with OA, is positive, and is looked on as due to rotation about an o,xis drawn from O upwards from the plane of the figure. Thus the rotation of the sun and the orbital motions of the Fig. 10. planets take place in the positive direction about axes drawn on the whole northwards from the plane of the ecliptic ; or we may put it thus : seizing an axis by the positive end, we must wwscrew by the negative end, we must screw to give positive rotation. And when, later, we consider rotations about three rectangular axes, O.r, Oy, Oz, we shall suppose them so drawn that rotation through a positive right angle about Ox changes Oy into Oz, Oy 02 O.r, Oz Ox Oy, the three letters being throughout arranged in cyclical order, xyz, yzx, &c.] 45. Here we must introduce a simple geometrical pro- Geometri- position : JJ*gj|j If any point be taken in the plane of a parallelogram, ^ and triangles be formed with the point as vertex and with contiguous sides and the conterminoiis diagonal as their respective bases, the sum of the areas of the first two triangles is equal to the area of the third. Thus, in areas (fig. 11), OAB+OAC=OAD. If He within the angle BAG, as in fig. 12, the propo sition becomes OAC-OAB=OAD. A A Fig. 11. Fig. 12. 46. Remembering that these areas represent half the
moments of the bases of the respective triangles about the