Proposition 34

Table of Contents

PROPOSITION 34 Problem

Let the uniform force acting be g pulling downwards everywhere and the given curve AT (Fig. 39) [p. 140] ; to find the curve AM, upon which a body thus descends, so that the time to pass through any arc AM is proportional to the square root of the corresponding applied line PT of the given curve AT.

Solution

The common abscissa AP = x, the applied line of the curve AT is PT = t; hence the equation between x and t is given, since the curve AT is given, which must be such that with x = 0 makes t = 0 also, since the initial motion is put at A and the times reckoned from the point A. Again let the curve sought be AM with the applied line PM = y and the arc AM = s. Let the initial speed at A correspond to the height b. Hence the speed at M corresponds to the height b + gx and the time in which the arc is completed is equal to which must be equal to t.

Hence this equation is obtained : ..

Hence .. and ..

From which equation, since t is given through x, the curve sought AM can be constructed. Moreover this has to be constructed so that x = 0 also gives y = 0, as the start of the curve AM is at A. Q.E.I

Corollary 1

Therefore in order that the curve is real, it is necessary that bdt 2 + gxdt 2 shall be greater than 4tdx 2 , or [p. 141]

For if the relation becomes then the curve AM becomes a vertical straight line, upon which the quickest descent is made.

Corollary 2

Therefore if the curve AT somewhere makes dt equal to 2 t dx , then the tangent b + gx there corresponding to the curve AM is vertical. And if beyond this point it satisfies dt < dx then the curve AM does not descent to that point, but has a turning point at 2 t b + gx that point where the tangent is vertical. Corollary 3.
If the angle that the curve AT makes to the vertical AP at A is acute, the tangent of which is m, then at the beginning A : hence m( b + gx ) must be greater than 4x, which always happens if b is not equal to 0. Moreover it then becomes : dy Therefore with x = 0 , dx =∝ , or in these cases the tangent to the curve AM at A is horizontal, unless b = 0. But if b = 0, then it becomes Therefore least the curve AM becomes imaginary, gm must be greater than 4 and then the curve AM makes an acute angle with AP at A, the tangent of which is ( gm − 4 )

Corollary 4

If the angle that the curve AT makes with the vertical AP at A is right, then m =∝ . Therefore in this case the tangent to the curve AM at A is always horizontal, then either b is made equal to zero, or otherwise.

Corollary 5

If the speed at A is equal to 0 and at the start A the curve AT is combined with the curve, the equation of which is t = αx n with the number n taken as positive, so that as x increases, so too does t, then we have :

Now least dy becomes imaginary on making x = 0, it must be true that n > 2n − 1 or n < 1, in which case clearly the curve AT is normal to AP at A. For now at the point A, n 2αgx n

and the radius of osculation of the curve AM at A is equal to 2( n −1 ) . From which it follows for the curve AM, the tangent of which is horizontal at A, that the radius of osculation at A must be infinitely small, if the body is able to start from rest on that curve.

For unless the radius of osculation is infinitely small, the body remains at rest at A forever.

Corollary 6

Therefore if the body placed at A descends from rest, so that the curve AM is made real, then dt must be greater than dx , even at the start of the curve AT. Whereby if we ..

put where p is a positive quantity, even if x is made exceedingly large, then we have : where ∫ pdx must thus be taken, so that it vanishes on putting x = 0. Moreover with this value in place, on substituting for dt , there is produced …

for the curve AM sought. Or this equation is obtained between x and y : dt = 1 + p into the above expression for dy

p cannot be such a quantity that ∫ pdx can be made infinitely large when integrated in the prescribed manner.

Corollary 7

From what has been said it is understood that as long as the value p is kept positive, the body descends the curve AM ; if we male p = 0 and then negative, then the curve has a cusp at that place and returns up again. If p =∝ with pdx yet remaining finite, then ∫ the curve AM has a horizontal tangent there.

Corollary 8

If b is not put equal to 0, from the same curve AT innumerable curves AM can be found; since the initial speed can indeed be taken as greater or less, with another curve AM produced.

Scholium

The greatest use of this problem is in the solutions of the following indeterminate problems, in which all the curves are required, upon which a body in the same time arrives at either a given straight or curved line. On account of this we will investigate the innate character of the quantities t and p with more care, so that we are allowed to use these in the following propositions.