Proposition 35

Table of Contents

PROPOSITION 35. Problem

With a uniform force acting in the downwards direction, to find all the curves AMC (Fig. 40); upon which a body beginning to descend from rest at A, arrives at the horizontal line BC in a given time.

Solution

Putting AP = x, PM = y and AB = a. In the curve AND, PN expresses the above assumed quantity pdx , and a property of this curve must be that it meets the axis AB at A, and that the applied lines increase on being continued even as far as D, so that clearly pdx is positive.

By taking (312) the time to traverse AMC = …

On account of … which, since an infinity of curves of this kind can be substituted in place of the curve AND, from these an infinite number of curves AMC can arise, upon all of which a body reaches the horizontal line BC in the same time from A [p. 145].

Therefore in order that this may be obtained, such a quantity must be taken for pdx , which vanishes when x = … 0 and which becomes equal to BD on putting x = a, with p retaining a positive value everywhere along AND. Q.E.I.

Corollary 1

If on making x = a, p = 0, or if the curve AND at D stands perpendicular to the horizontal CD, then the curve AMC also remains perpendicular to DC.

Corollary 2

If on putting x = 0 also p = 0, the tangent to the curve AMC is vertical at A ; now likewise also it comes about if p x becomes equal to zero on putting x = 0. But if p x becomes infinite on putting x = 0, then the curve AMC has a horizontal tangent at A.

Scholium 1

Therefore the problem is understood to be indeterminate to a large extent, since in an infinite number of ways an infinite number of curves AMC can be found. On account of which in the following examples we indicate however many series of the infinite curves sought satisfying the question that we have been pleased to find.

Example 1

Putting PN = pdx = z and BD = b , thus in order that the descent time must be equal to …

For the curve AND, this equation is taken: z = αx 2 + βx , which ..

now has this property, as pdx or z vanishes on putting x = 0. Now since on making x = a we must have z = b , there is obtained b = αa 2 + β a and hence β = ab − αa and thus z = αx 2 + x a b − αax.

dz must always have a positive value, if x < a, it is necessary that

Then since p or dx .. 2αx + ab − αa is positive. Whereby it is required that putting b = αa 2 + αaf , then α = … z= b > αa 2 ; therefore on …

With which substituted, there is obtained .. a 2 + af x 2 b + fx b , which equation, with innumerable positive values substituted in place of a 2 + af f , gives the curves AND.

Moreover there becomes : from which it is apparent that all the curves AMC hence arising are tangents to the line AB at A. Now the equation for the curves AMC is this [from (312)] : Which contains an infinitely of curves satisfying the problem, upon which the descent time for all to the horizontal line is equal to …

Corollary 3

Now all these line are rectifiable. For since then we have ..

Thus the total area under the curve AMC is equal to : …

Corollary 4

Therefore among these curves AMC the longest is produced if f = 0; for then AMC = a + 54 gab. And for this, that equation becomes :

The shortest is obtained by making f =∝ ; for then it becomes AMC = a + 23 gab.

The equation for this curve is :

Scholium 2.

All the curves AND contained under the equation are parabolas, thus so that by parabolas alone innumerable curves are found satisfying the problem. Now neither are all the parabolas contained by this equation, as in place of this equation, if this other equation is used …

which also contains an infinite number of parabolas, again an infinite number of curves AMC are found, upon which a body completes the descent in the given time. From which it is to be understood that if only conic sections are substituted in place of the curve AND, then such an infinite number of curves AMC can be found.

For taking this equation for the curve AND : z 2 + αz = β x 2 + γx + δxz , which contains all the conic sections passing through the point A, then it must become [for BD]: … b + α b = βa 2 + γa + δa b γ and both α and β a + γ +δ b must be positive quantities; which is easily seen, since it can α + 2 b −δa be done in an infinite number of ways. If then all the algebraic curves are considered and afterwards also the transcendental curves likewise, the greatest wealth of all the curves described in the same way con be conceived.

Example 2

This general equation z = x n b is taken for the curve AND with n denoting some … positive number; z vanishes on putting x = 0 and the equation becomes z = b on putting … x = a, as it is requires; now besides also the quantity p or dx is positive.

Therefore since [the appropriate quantity] becomes p gx = … then the equation for y is :

Which equation includes an infinite number of curves AMC, which are all rectifiable.

For the arc becomes : … and thus ….

Corollary 5

If n = 12 , then the equation … becomes .. and …

Whereby the curve becomes an inclined straight line, upon which the descent is in a time … 2 ga equal to g + b . Therefore it is evident the shorter lines [i.e. curves] are given by this inclined straight line, upon which a body from A arrives at the horizontal BC in a given time; indeed on making n < 12 the line AMC becomes shorter.

Scholium 3

Otherwise if a single curve AND is given, and it is required to provide a curve AMC, from that curve itself innumerable others can be found. For with one equation given between z and x taking hence with diverse values of m, innumerable curves are found. In a similar manner also we can put : for it becomes PN = z = b , on putting x = a. And in general, if P is any function of x and z, A is now the same function that is produced on making x = a and z = b , and it can be taken that PN = Pz . Moreover P must be such a function that Pz vanishes on A making x = 0 and z = 0, and the differential of PN divided by dx must be a positive quantity, for as long as x < a.

Scholium 4

In a similar manner the most general problem can be solved, if on designating P as some function of x vanishing if x = 0, and A is that quantity which P becomes if x = a, on taking z = P Ab , for the most general equation for the curve AND. Then we have Q b dP = Qdx, Q must be a positive quantity, as long as x is not greater than a ; then p = A and hence which is the most general equation for the curves AMC, which are all completed by a body descending in a proposed time. It is apparent in this way that transcendental curves can also be substituted in place of the curves AND, in which cases the time, in which some part of the curve AMC is completed, cannot be defined algebraically. If Q gbx is put equal to R, then we have

Therefore with some function of x taken in place of R in order to find A, … Rdx must be gbx … integrated, so that it thus vanishes on putting x = 0; then it is required to put x = a, and as that comes about, it is equal to A. Now here only this has advise has to be given, that a positive quantity is taken for R, for as long as x does not exceed a, and it must be warned that Rdx should not becomes infinite, if the integral is taken in the prescribed manner.