Proposition 94

PROPOSITION 94. PROBLEM

771. If the body is moving on the curve AMB (Fig. 71) in whatever manner, while the curve itself is revolving around the central fixed point C, it is required to find two forces, one of which is always acting towards the fixed point C, and the other is directed normally to the line in the given position PC, which two forces have the effect that the body is free to move in this orbit.

SOLUTION

Let the speed of the body present at M, in which it traverses the element of the curve itself Mm, correspond to the height v, and the angular speed of the body in the orbit to the true angular speed of the body be in the ratio about C as 1 to w, or the angular speed of the body in orbit to the angular speed of the orbit itself, while the body is at M, is as 1 to w –

1. [If ωa , ωo , and ωb / o are the absolute, orbital, and relative to the orbital, angular speeds of the body then ωa = ωo + ωb / o ; Euler sets …

The radius … is put equal to y and the perpendicular CT from C sent to the tangent of the orbit at M is equal to p, while the tangent itself MT is equal to q, thus in order that q = ( y 2 − p 2 ) . From M the perpendicular MP which is called z, is sent to the position of the given line DP, and CP is called x, thus in order that x = ( y 2 − z 2 ) . Now while the element Mm is traversed, meanwhile the orbital angle is put equal to mCμ of the circumference; as from the composite motion of the body it arrives at μ, with Cμ = Cm , Mμ is an element of the true curve in which the body is moving, in which produced the perpendicular CΘ is sent from C. With centre C the little arc Mnν is described, [p. 321] where we have : mn = μν = dy et Mn : Mv = 1 : w .

Truly, we have ydy Mm = q

hence Mv = wpdy , q from which is found : and again, [as CΘM and Mμν are similar triangles :] and The ratio is made Mm : Mμ = v : v( w 2 p 2 + q 2 ) v( w 2 p 2 + q 2 ) , the square of which shows y y2 the height corresponding the true speed of the body ; therefore the increment of this is equal to Moreover, the radius of osculation of the true curve, in which the body falls, is equal to : With the whole sine put as 1, the sine of the angle CMP is equal to xy and the cosine is equal to zy . But the sine of the angle CMΘ is equal to is equal to …

and the cosine of this …

From which the sine of the angle PMΘ is equal to and the cosine of this is equal to wpx + qz y ( w2 p 2 + q 2 ) qpz − qz y ( w2 p 2 + q 2 ) .

By sending the perpendicular PQ from P to the tangent MΘ , then

And with the body traversing the element Mν the increment of the line PM is From which equation the relation between w and x becomes known and likewise the position of the apsidal line AB with respect to the line CP can be found.

Now put the force acting on the body towards MC equal to P and the force pulling along MP equal to Q, [p. 322] from which there arises the tangential force retarding the motion of the body, which is equal to : dy which therefore taken by q ( w2 p 2 + q 2 ) can be put equal to : thus in order to give :

Moreover, the normal force arising from both is equal to : which must be equal to : (561), whereby we have :

From which equations solved for P and Q, we have : and

Truly the angle, that the apsis line AB makes with the line CP, is equal to thus the position of this is known for any time. Q.E.I.

Corollary 1

Since dz = , putting z = ty then there comes about qy From which equation, if w is given in terms of y, in which also on account of the given curve AMB, p and q are expressed, t can be found and likewise also z and x. Corollary 2. 773. If the speed of the body in the orbit v varies inversely as the perpendicular CT 2 sent from C to the tangent or v = a 2c , then we can write : p

Corollary 3

774. If in this case w is constant, the force Q vanishes and there remains alone pulling towards the centre C, which effects the motion, as the body progresses in the orbit AMB moving around C, evidently as has been found above (734)

Corollary 4

775. If v is not a 2c , but w is constant, thus as the angular motion of the orbit is p proportional to the angular motion of the body in the orbit truly as w – 1 to 1, then we haveE

Example

776. On putting v = a 2c the curve AMB is an ellipse having C in either focus. Therefore p if the transverse axis of this ellipse is called A and the latus rectum L, then Whereby we have

Scholion 1

777. These formulae for the curve of the ellipse can be made simpler in various ways, if the curve in which the body is moving is approximately circular. And in this case it is of some use in the theoretical motion of the moon to be defined. [p. 324] For the earth is put at rest at C and the sun on the line CP perpendicular at C is considered equally as being at rest ; with which put in place and with these forces compared both with the forces of the sun and the earth, the synodal motion of the moon is elicited for some position of the apsidal line and likewise the motion of the apsidal line, which only differs slightly from the true motion of the moon.

Scholium 2

778. This proposition certainly appears of greater extent than the above (729), in which all the force was directed towards the centre of rotation of the orbit; indeed the former is included with the force Q vanishing. Yet the quadrature cannot account perfectly for the motion of the moon because the proportional force in P varies inversely with the cube of the distance MC (773). Because of this we offer other orbits besides the gyrations in the middle, and which appear wider and agree more with the physics questions. Of this kind are the motions of orbits by which curves are always in orbits parallel to themselves, which on contemplation deserve to be preferred from others, since the forces acting are both easier to find and the formulas are simpler to understand. Moreover there is an outstanding need for this, and the following theorem is presented.