188 REPORT—1862. 
by adding to the central force required to make the body move in the same 
orbit at rest, a force q (dist.)-3. This appears very readily by means of the 
differential equation (antée, No. 6), viz. writing therein P+-cu’ for P, and then 
6', 2! in the place of o/1—S, 
its original form, with 6!, h’, in the place of 6, 2 respectively. 
12, It may be remarked that when the original central force vanishes, the 
fixed orbit is a right line (not passing through the centre of force). It thus 
appears by § 9 that the curve u=A cos (n6+B) may be described under the 
action of a force q (dist.)-3. A proposition in § 2, already referred to, shows 
that a logarithmic spiral may be described under the action of such a force. 
13. But the case of a force & (dist.)—3 was first completely discussed by 
Cotes in the ‘ Harmonia Mensurarum,’ pp. 31-35, 98-104, and Notes, pp. 117 
-173. There are in all five cases, according as the 
velocity of projection is 
1. Less than that acquired in falling from infi- 
nity, or say equal to that acquired in fall- 
ing from a point C to P, the point of pro- 
jection. 
2, Equal to that acquired in falling from infi- 
w/ 1— a respectively, the equation retains 
nity. 
3, 4, 5. eats than that acquired in falling 
from infinity, or say equal to that acquired 
in falling from a point C’, through infinity, 
to P; viz. PQ being the direction of pro- 
jection,and SQ, C'T perpendiculars thereon 
from § and C' respectively, 
3. SQ<TQ; 
5. SQ>TQ; 
the equations of the orbits being 
1. w=Ae™+4+Be~™, A and B same sign, so that rad. vector is never infinite. 
2. u=Ae” or Be~™, logarithmic spiral. 
3. u=Ac™+Be7-™, A and B opposite signs, so that rad. ector becomes 
infinite. 
4. u=A0+B, m=0, reciprocal spiral. 
5. u=A cos (n0+B), m=ny —1. 
14, The before-mentioned equation, 
Cu P 
apt — ee 
0, 
is in effect given (but the equation is encumbered with a tangential force) in 
Clairaut’s “ Théorie de la Lune,” 1765. Itis given in its actual form, and ex- 
tensively used (in particular for obtaining the above-mentioned equations for 
Cotes’s spirals) in Whewell’s ‘ Dynamics,’ 1823. The equation appears to be 
but little known to continental writers, and (under the form wv" +u—a’r?>R=0) 
it is given as new by Schellbach as late as 1853. The formule used in place 
of it are those which give ¢ and @ each of them in terms of r; viz. 
? 
, 
r 
; 
