PROPOSITION 91. Problem

PROPOSITION 91. Problem

833. In any given surface to determine the line, that the body describes in that motion, acted on by no forces either in a vacuum or in a medium with some kind of resistance.

Solution

Because the body put in place is not acted on by any absolute forces, the line described by that on the surface is the shortest line in vacuo (62). But the force of resistance in a medium only diminishes the speed of the body and does not affect the direction in any manner; whereby also in a medium with resistance the path described by the body on some surface is equally the shortest. Therefore with the variables in place as before : AP = x, PQ = y and QM = z, (Fig. 91) let dz = Pdx + Qdy be the equation expressing the nature of the surface and Mm , mμ any two elements of the shortest line. From these found above (69) for the shortest line, this is the equation:

Thus there arises :

But the equation for the surface differentiated gives : with these connected together there is given

Therefore with the element Mm given then the following element mμ on the shortest line is found; for it is given by : and the values of ddy and ddz have been found. Whereby hence the position of any following element is determined and the nature of the shortest line by some projection of these is known. Q.E.I.

Corollary 1

834. If, in the equation for the surface P and Q are given in terms of x and y only, then the equation denotes the projection of the shortest line in the plane APQ.

Corollary 2.

835. Therefore for the shortest line Mmμ , with the elements selected equally from the axis, then : and from which equations the point μ is known from the two preceding points M and m.

Corollary 3

836. Because the angle RMN vanishes for the shortest line (71), R falls on N; hence the position of the radius of osculation thus is obtained, in order that AX = x + Pz and XR = −Qz − y . Now the length of the radius of osculation (73) is equal to

Corollary 4

837. Now the plane IMR, in which the shortest elements Mmμ are situated, is thus determined, as it becomes : and Now the tangent of the angle, that the plane IMR makes with the plane APQ, is equal to :

The secant of this angle is equal to : or the cosine is equal to :

Example 1. [p. 467]

838. Let some cylindrical surface have the axis AP; the nature of this is expressed by the equation dz = Qdy with P vanishing in the general equation dz = Pdx + Qdy . Whereby for the projection of the shortest line of this surface in the plane APQ on account of P = 0 and dP = 0 there is obtained this equation: or if indeed Q is only given in terms of y ; but if Q is given in terms of y and z, the variable can be eliminated with the help of the equation dz = Qdy . As in the circular cylinder, in which z 2 + y 2 = a 2 , then Whereby it follows that …

∫ Moreover in general, dy (1 + Q 2 ) expresses the arc of the section normal to the axis AP; whereby with the said arc equal to s then αx = s . From which it is understood, if such a surface is set out on a plane, to be the line of the shortest straight line, as agreed.

Example 2.

839. Let the proposed surface be some cone having the vertex at A; the equation for such a surface thus can be adapted, so that z is equal to a function of one dimension of x and y. Whereby in the equation dz = Pdx + Qdy the letters P and Q are functions of zero dimensions of x and y. On this account, as now shown elsewhere, it follows that [see E044] : hence [on differentiation] there becomes : [p. 468] and and finally : With which substituted, there is : Put y = px ; P is equal to a certain function of p only, because P is a function of zero dimensions of x and y. Now there is : and

From which equation indeed the projection can hardly be recognized. Moreover how the shortest line in such a surface is to be determined, I have set out in more detail in Comment. III. p. 120, [E09 in this series of translations]. Moreover the same as before is to be noted concerning the shortest line, clearly because that set out from the conical surface becomes a straight line in the plane.

Scholium.

840. I will not tarry here with the determination in a similar manner of the shortest lines on other forms of surfaces, since in the place cited I have set out this material more fully. Hence I progress to the investigation of the lines which are described on a surface by a body acted on by some forces. Now before this, it is necessary that we examine more carefully the effect of each force.

Definition 4

841. In the following we call the pressing force that normal force, the direction of which is normal to the surface itself in which the body is moving.

Corollary

842. Therefore this pressing force either increases or decreases the centrifugal force, according as the direction of this force falls either opposite to the direction of the radius of osculation of the shortest line, or in that direction (79).

Definition 5

843. In the following we call the force of deflection that normal force, the direction of which is on surface in the tangent plane, and perpendicular to the path described by the body.

Corollary.

844. Hence this force deflects the body from the shortest line that the body describes when acted on by no forces, and either draws the body to this or that side [of this line] as the direction of this force either pulls the body this way or that.