Proposition 37

Table of Contents

PROPOSITION 37. PROBLEM

To determine the time of descent through AC (Fig. 28) to the centre of force C, if the centripetal force varies inversely with the reciprocal of the distance, the exponent of which is raised to the power mm−1 , with m denoting some positive integer.

SOLUTION

…

Therefore the element ….

Thus, we set n = 1−mm and since v = …

of time , that is dy , is equal to dy : …

and the time to descend .. through PC is equal to :

…

Putting a m = b and y m = z ,then dy = mz m −1dz; and hence the time to pass through PC is equal to :

…

But for the integral of ..

, in the same manner as taken in the preceding Prop., we (b − z ) have : 2b 2 (1 − ( m −1) + ( m −1)(m − 2) − ( m −1)(m − 2)(m −3) + etc.) . ..

On account of which the time for the descent through AC, with a 2 m in place of b 2 , …

becomes equal to 2 ma m f m multiplied by this series :

…

Thus as the amount m is a positive integer, so the series total is finite, in order that the time sought can thus be expressed algebraically. E. I.

Corollary 1

Let m = 1, in which case n = 0, and the centripetal force is uniform and therefore equal to gravity. Hence the series is equal to 1, and the time to fall through AC = 2 a , as everything has now been found as in §219 with the letter m ignored. [p. 122]

Corollary 2

Let m = 2, as now n = 2 ; then the time to fall is 3 .2a f

Let m = 3, as now n = −32 ; and the whole time to descent is equal to

In a similar manner, if m = 4 and on this account, n = −43 , and the time to fall produced

is equal to : 32..54..76 .2a f

Corollary 3

Generally therefore for whatever m shall be, and thus n = 1−mm ; the time to fall the whole distance AC …

Corollary 4

With the same interpolations used as above (294) the times of descent can be assigned, if m is any positive integer + 12 .

For m = 12 , in which case n = 1; the time of descent is equal to π2 2 f , in short as in § 283, where the same case, in which n = 1 or the centripetal force is proportional to the distance, has been explored.

Corollary 5

If m = 32 , or n = −31 , the time of descent is equal to 12 . π2 6 .a f or n = −53 , the time of descent produced is 12 . 34 . π2 in which n = 1−mm , the descent time is found

In the general case,

Scholium

From these it is understood, that for whatever cases the times of descent can be expressed algebraically, where n = −22mm−−11 or n = 1−mm and m specifies some positive integer.

Besides these cases I doubt that any other is given. Then the cases also appear in which the times depend on the quadrature of the circle, and these occur, if either

…

with m denoting as above positive integer. Indeed nor are these all the cases which can be deduced from the quadrature of the circle, for there is the singular case, for which n = –1, which depends on the quadrature of the circle too, as we will show in the following proposition.

For this is a different case from these, since here in the expression for the time not π but π occurs; and besides also only the whole time of descent can be shown involving π , since the time for any indefinite interval can be shown except for the quadrature of the whole transcendental curve