ON THE SPECIAL PROBLEMS OF DYNAMICS. 187 
Central Forces, Article Nos. 6 to 26. 
6. The theory of the motion of a body under the action of a given central’ 
force was first established in the ‘ Principia,’ Book I. §$ 2 & 3: viz. Prop. I. 
the areas are proportional to the times, that is (using the ordinary analytical 
1 hy 
notation), °d@=hdt, and Prop. VI. Cor. 3, Pa Sy py ate apt") so 
that : Mu P 
a3 tu—a>5=0. 
do hew 
7. It,is to be noticed that, given the orbit, the law of force is at once 
determined ; and § 2 contains several instances of such determination ; thus, 
Prop. VII. If a body revolve in a circle, the law of force to a point § is 
a 
force Agp: py (P the body, PV the chord through §). 
Prop. IX. If a body move in a logarithmic spiral, force q (dist.)-3. 
Prop. X. Ifa body move in an ellipse, force to centre @ dist., and as a parti- 
cular case, if the body move in a parabola under the action of a force 
parallel to the axis, the forcé is constant. The particular case of motion in 
a parabola had been obtained by Galileo. 
And § 3. Props. XI. XII. XIII. Ifa body move in an ellipse, hyperbola, or 
parabola under the action of a force tending to the focus, force q@ (dist.)—2. 
8. But Newton had no direct method of solving the inverse problem 
(which depends on the solution of the differential equation), ‘Given the 
force to find the orbit.” Thus force q& (dist.)—2, after it has been shown that 
an ellipse, a hyperbola, and a parabola may each of them be described under 
the action of such a force. The remainder of the solution consists in showing 
that, given the initial circumstances of the motion, a conic section (ellipse, 
parabola, or hyperbola, as the case may be) can be constructed, passing through 
the point of projection, having its tangent in the direction of the initial 
motion, and such that the velocity of the body describing the conic section 
under the action of the given central force is equal to the velocity of pro- 
jection ; which being so, the orbit will be the conic section so constructed. 
This is what is done, Prop. XVII. ; it may be observed that the latus rectum 
is constructed not very elegantly by means of the latus rectum of an 
auxiliary orbit. 
_ 9. A more elegant construction was obtained by Cotes (see the ‘ Harmonia 
Mensurarum,’ pp. 103-105, and demonstration from the author’s papers in 
the Notes by R. Smith, pp. 124, 125); depending on the position of a point C, 
such that the velocity acquired in falling under the action of the central 
force from C directly or through infinity* to P the point of projection, is equal 
to the given velocity of projection. as 
10. But Newton’s original construction is now usually replaced by a con- 
struction which employs the space due to the velocity of projection considered 
as produced by a constant force equal to the central force at the point of pro- 
jection. st 
; Al. Section 9 of Book I. relates to revolving orbits, viz., it is shown that 
a body may be made to move in an orbit revolving round the centre of force, 
* Tn the second case C lies on the radius vector produced beyond the centre, and the 
body is supposed to fall from rest at C (under the action of the central force considered as 
repulsive) to infinity, and then from the opposite infinity (with an initial velocity equal 
to the velocity so acquired) to P. 
