This gives the distance of the vertex V from the line Ax, i. e. shews the greatest positive distance of the moving point from Ax.

45. If L be the latus rectum of the parabola

This might have been obtained in the following manner:

If SY be perpendicular to AC,

46. The distance from A of the other point where the para

This value might have been obtained by considering that

the time in question = 2. time to vertex = 2.

v' sin

and the

a

distance from A of the other point where the parabola cuts Ax

The directrix of the parabola will be parallel to Ax, and will

cut Ay at a point E whose distance from A = AS=

212
2α

47. The velocity at Pin direction Ax will be v'cos, and that in direction Ay will =v'sin - at; therefore if v be this velocity,

A. 48. Both of the foregoing cases of motion may be investigated by the help of the differential calculus, as follows:

(I) If the acceleration be in the direction of motion, we have

If s is measured from the initial position of the moving point, then will = 0, for s and t vanish together. If the moving C point be initially at rest, then v'= 0.

A. 49. (II) If the acceleration be not in the direction of motion we have (using the figure and notation of Art. 42)

(No constants are added in the last integrations,

because x, y, and t all vanish together.)

y are

v'2

cos2

=

2α

α

.. (x - sin 24)" - 2000 (sin' - y),

し

which shews that the path is a parabola, the co-ordinates of

negative direction of y, i. e. in the direction of the acceleration;

A. CHAPTER IV.*

OF THE MOTION OF A POINT AFFECTED BY AN ACCELERATION, THE DIRECTION OF WHICH ALWAYS PASSES THROUGH A FIXED POINT.

50. THIS will evidently be of two kinds, according as the initial direction of motion does or does not pass through the fixed point.

(I) Let the initial direction of motion pass through the fixed point, then it is clear that the path is a straight line.

We shall, for simplicity's sake, consider the moving point to be initially at rest at a given position in space.

(j) If the acceleration be constant, the notion is that determined in Arts. 38—41.

51. (ij) Let the acceleration vary as the distance from the fixed point, its direction being towards it.

Take the fixed point for the origin of distance; and let a be the initial distance of the moving point, s its distance, and v its velocity at a time t, reckoned from the beginning of the motion.

The acceleration will be expressed by As, where λ = the measure of the acceleration at distance unity†. The direction of this is from the moving point to the fixed point; therefore if we

* Investigations of the results of the Articles in this Chapter, obtained without recourse to the Differential Calculus, will be found in Newton: in all cases the words "force" and "body" should be replaced by "acceleration" and "point".

+ N. B. As λs (the acceleration) is of 1 dimension in space and -2 in time, must be of no dimensions in space and −2 in time: we should more properly have said that the acceleration at distance unity is measured by λ times the unit of space.

consider the first part of the motion, which is clearly towards

d

2

.*. 8=

dt

or

dt2

+x)"

+λ) .0= A cos (√.t+B),

where A and B are constants, and have to be determined.

We have by differentiation

ds

or v=

dt

-. A sin (√.t+B).

Then initially t = 0, s = a, v = 0;

.. B=0, A= a;

and we have sa cos (√.t),

v = ÷√. a sin (√ł. t).