Proposition 25

PROPOSITION 25. Problem.

224. If a body is drawn downwards by some constant force, to find the curve AM (Fig. 32), that a body descending on that curve presses upon equally everywhere.

Solution.

Let AM be the curve sought ; with the abscissa to the vertical called AP = x, the applied line PM = y and the curve AM = s. Again let the force acting on the body at M be equal to P and the height corresponding to the speed at A = b; [p. 98] the height corresponding to the speed at M = b + Pdx , with the integral Pdx thus …

taken in order that it vanishes on making x = 0. With these in place the compression force that the curve sustains along the normal MN is equal to (83) with the element dx taken as constant [i. e. x is the independent variable].

Since this force has to be constant, it is put equal to k, then we have : But if ds is made constant, then we have ..

[which amounts to the starting condition where k = ds , as the centrifugal effect on the curve is zero when y is incremental initially. The different ways of expressing the radius of curvature should also be noted : In this case, if ds is constant, then from ds 2 = dx 2 + dy 2 we have 0 = dxddx + dyddy , and in an obvious notation, we have … from which − sin θdθ = ddx ….

Hence dθ = dxds × sin1 θ = dxds × dy. Hence R = ddsθ = dsdx …

is the required radius of curvature.] the integral of which is :

This [differential] equation can be constructed, as P is given in terms of x, and since y is not present in this equation, but only dy. Q.E.I.

Corollary 1

225. The integral expresses the time in which the body starts from A, with the same initial speed as that with which it is moved along AM, and falls straight down through the height AP, and ∫ ( b + Pdx ) gives the speed at the same place. Whereby this speed at P divided by the time to traverse AP gives 2kds , from which property the curve AM can be determined.

[Thus the quantity 2kds , related to the angle of projection at A, is determined from special dy values of the time and speed in the above equation, which is used further.]

Corollary 2

226. Moreover the time to traverse the distance AP can be increased by any constant quantity, for example c , [i. e. in the ratio b ]. And from this constant magnitude the c angle which the curve at A makes with AP is determined. [The sine of the angle between dy the vertical axis and the curve at A is of course ds , and 2kds … thus, increasing c

makes the angle greater, up to a certain allowable value ] Clearly the sine of the angle is equal to k c with the total sine put equal to 1. Whereby c cannot thus be taken greater … than 2 k b , and if the body starts at rest from A, then c must be equal to 0.

Example. [p. 99]

227. Let the force be uniform or P = g; then we have : …

[Note that the time has been increased by the added constant 2kg c ] hence we have : …

From which the following equation arises, [noting that dx …] then …

Therefore with these substituted, we have : …

This equation allows the integration for three cases, the first of which occurs for k = 0; for then the curve is found that a body describes freely projected from A. The second case is …

when h = 0 or b = c ; for then we have ds = kg or the line satisfying the equation is an inclined straight line. If in the third case k = g or the total compression force is everywhere equal to the force g acting on the body, then the equation becomes :

the integral of this is : …

This constant, since with x = 0 or t = b makes y = 0, must be equal to : Therefore with ( b + gx ) restored in place of t and by placing found that :

b − c = h = a , it is

..

Which is the equation of the curve sought, in which a must be a number less than b.

If the body must fall from rest, any other line besides a straight line is not satisfactory.

It must be the case that c = 0, in order that the angle at A is real, and therefore the equation is put in place :

Corollary 3

228. An algebraic equation found, if it is to be free from irrationality, can be made of order five. If a = b is put in this equation, in which case the tangent of the curve at A is vertical, and gives :

Corollary 4

229. If in general the tangent at A is to be vertical, then c = 0 and thus this equation arises :

If the tangent at A is placed horizontal, then k c = g b and this equation arises :

Scholium

230. This curve is called the line of uniform pressing [ above we have called this the compression force, and of course it refers to the normal reaction force exerted on the body by the fixed curve having constant magnitude, although the reaction of this force is usually considered in the text;

It is inappropriate to use modern terms, while ‘pressing’ or ‘squeezing together’ seems too vague. Again, the derived word ‘pressure’ has a different meaning now, and cannot be used. Remember that many of the latin words recruited by Euler are given different mathematical meanings from those originally found in the dictionary. Even the word ’expression’ seems to relate to extracting the juice of the grape. All part of the fun of being Euler, I suppose, and although he was a person of great piety - see some of his Letters to a German Princess, he nevertheless had a sense of humour.] and the solution of this problem is set out in the Comment. Acad. Paris., which agrees uncommonly well with our solution.

(G. F. De L’Hospital (1661 - 1704), Solution d’un problème physico-mathématique, proposé par Iean Bernoulli, Mém. de l’acad. d. sc. de Paris 1700, p. 9. See also the study by P. Varignon (1654 - 1722), Usage d’une intégrale donnée par G. F. De L’Hospital, ou sur les pressions des courbes en général, avec la solution de quelques autres questiones approchantes de la sienne, Mém. de l’acad. d. sc. de Paris 1710, p. 158. References by P. S.)

The other case agrees with this solution, if the force is not constant, but for whatever the variable P, the equation found is nevertheless integrable, if the compression on the curve should be proportional to P. [p. 101] For it becomes k = mP and the following equation is produced for the curve sought :

…

the integral of which is :

…

This equation, if c = 0, is for a straight line inclined to the horizontal. But the angle is defined by c , which the curve makes with the vertical at A; indeed the sine of this angle is m + m c . Whereby if we take …, the curve is a tangent to the vertical at A.

Besides, this curve has the property that the time in which the arc AM is traversed is proportional to m • AM − PM . Finally from the solution of this proposition flows the solution of the following, in which from the given curve and the equal compression of the curve, the magnitude of the force acting downwards is sought.