Chapter 6u

Proposition 128

Euler
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PROPOSITION 131. PROBLEM

  1. If a body M (Fig.97) in some resisting medium is drawn by three forces, of which the direction of one is Mf parallel to the AP, the direction of another Mg is parallel to the applied line PQ placed in the plane APQ and the direction of the third is MQ sent normally to the plane APQ from M, to find the motion of the body and the line that it describes.

SOLUTION

As before by putting AP = x. PQ = y and QM = z and with the speed at M corresponding to the height v, let the force drawing along Mf be equal to P, the force drawing along Mg be equal to Q, and the force drawing along MQ equal to R, and the force of the resistance at M is equal to V. These three forces can be resolved into three others, the directions of which agree with these in the previous proposition, [p. 475] and the tangential force produced (these are the normal forces) and (823).

For here we use the same denominators, as with these in Proposition 99. Therefore with these values substituted in the preceding formulas we have the following three equations

…

and Which three equations with v eliminated give two equations in the coordinates x, y and z, which express the nature of the curve described. Moreover in these formulae the element dx has been assumed constant. Q.E.I.

Corollary 1

  1. The two last equations agree perfectly with these that we found for the vacuum (823). Whereby the equations which follow from these have a place, as in the vacuum case so in the resistive case. Moreover the whole distinction that lies between the motion in the vacuum case and the motion with resistance, depends on the first equation.

Corollary 2

  1. Moreover from the final two equations solved together there arises this ratio : On account of which in place of second normal equation, which makes up the greater part, this substitution can be made [p. 476] or which does not involve v.

Corollary 3

1105a. With the help of this ratio the determination of the plane RMS is found in terms of first order differentials as follows : the tangent of the angle POR is equal to … and the tangent of the angle of inclination of the plane RMS to the fixed plane RQS is equal to (825).

Corollary 4

  1. If two of the forces P, Q, R vanish, the motion by necessity becomes that in a plane. For if P and Q vanish, this makes ddy = 0 ; if P and R vanish, this makes (if Q and R vanish, this makes ddz = 0;) dzddy = dyddz or dz = αdy. Which all indicate that the motion lies in a plane.

Corollary 5

  1. If P, Q and R are proportionals of x, y et z, then the body is always attracted to the point A, and thus the motion of this body becomes that in a plane. The formulae indicate the same; for set AO = 0. But because then it follows that xddy = xdzxddz and on integrating xdy − ydx = αxdz − αzdx. xdy − ydx − zdx Whereby αddz = ddy , thus the proposal is agreed upon. [p. 477]

Corollary 6

  1. If the force P vanishes, then the ratio becomes ddy : ddz = Q : R and With these values P, Q and R in place in the equation, by which dv is defined, on substitution there arises Where, if the resistance V is put equal to vc and on placing equal to ds, there becomes ( dx 2 + dy 2 + dz 2 ) , or Mm

Corollary 7

  1. If the force R vanishes, the ratio becomes and Therefore we have Where, if V = vc , there becomes In a like manner, if Q vanishes, there is produced :

Scholium

  1. All the forces can be reduced to these three forces P, Q and R, in whatever way they are able to be devised. On account of which, whatever problem that is proposed, two equations can be elicited that contain the nature of the described curve[p. 478]. Truly of these one is a differential equation of the second degree, and the other a differential equation of the third degree, if indeed the value of v found from the equation is differentiated and with the differential is substituted in place of dv in the equation

PROPOSITION 132. PROBLEM

  1. In a uniform medium, which resists in the simple ratio of the speeds, the body is always attracted normally to the line AP (Fig.97); to define the curve that the body describes projected in any manner.

SOLUTION

As before these are put in place : AP = x, PQ = y, QM = z, the speed at M = v the exponent of the resistance is equal to c, the force by which the body at M is drawn along MP , = S. With these in place, the resistance is given by hence the ratio Q:R = y: z. On account of which we have ddy:ddz = y:z and

The integral of this equation is ydz − zdy = αdx. Again also, as P = 0 we have this equation : (1108) on placing ds = ( dx 2 + dy 2 + dz 2 ) . The integral of this is With this value substituted, there is produced : Putting z = py; [p. 479] we have the following two equations, from which the nature of the curves described ought to be determined,

Q.E.I

Corollary 1

ds ( b − x ), the element of time 1112. Since it is the case that 2 cv = dx ∫ dsv = b− x . 2dx c

Therefore the whole time, in which the body is moved horizontally along AP by the motion, is equal to 2 c l b −b x . Therefore the horizontal motion agrees with the motion in the same resisting medium along the line AP with no force acting, with the initial speed at A corresponding to the height bb : 4c.

Corollary 2

  1. Truly neither does the motion have this special amount of time only in place if the body is drawn along MP or if Q : R = y : z, but it always prevails if P = 0. For this follows from (1108), in which P is put equal to zero.

Corollary 3

  1. Therefore the progressive motion of the body along AP has been slowed down, and it cannot go beyond the limit, which is x = b. Moreover the time taken is infinitely great for the body to be able to reach this limit. [p. 480]

Example

  1. We put the force by which the body is attracted to the line AP to be in proportion to the distances MP or

Therefore in order to determine the curve we have these equations in that equation is put y = e ∫ udx and it becomes q Which equation becomes separable on putting u = b − x ; for it produces Therefore with q given and on also on account of u given in terms of x. Consequently also y in terms of x is known, from which the projection of the curve described in the plane APQ is obtained. Then from the given y in terms of x, also p is given in terms of x on account of dp = αdx 2 , and likewise z in terms of x. On account of which the whole y curve described by the body can be constructed.

  1. If b vanishes, likewise also the progressive motion of the body along AP vanishes and on account of this the body moves in the plane through A normally to AP and is attracted to A in the ratio of the distances. Moreover the curve, which the body describes in this case, can also be constructed (1027) along with the others.

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