190 REPORT—1862, 
= sin 3@=3A pie, sin} g'=3q JU 
The form of the formula is, it will be observed, similar to that for motion in 
a straight line (anié, No. 4), and in fact the motion in the ellipse is, by an 
ingenious geometrical transformation, made to depend upon that in the 
straight line. The geometrical theorems upon which the transformation 
depends are stated, Cayley “On Lambert’s Theorem &c.” (1861). 
20. The theorem was also obtained by Lagrange in the memoir “ Re- 
cherches &c.” (1767) as a corollary to his solution of the problem of two 
centres; viz. upon making the attractive force of one of the centres equal to 
zero, and assuming that such centre is situate on the curve, the expression for 
the time presents ‘itself in the form given by Lambert’s theorem, 
21. Two other demonstrations of the theorem are given by Lagrange in 
the memoir “Sur une maniére particuliére d’exprimer Te temps &e.”’ (1778), 
reproduced in Note V. of the second volume of the last edition (Bertrand’s) of 
the ‘Mécanique Analytique.’ As M. Bertrand remarks, these demonstrations 
are very complete, very elegant, and very natural, assuming that the theorem 
is known beforehand. 
Demonstrations were also given by Gauss, ‘‘ Theoria Motus ” (1809), p. 119 
et seq.; Pagani, « Démonstration @un théoréme &e.” (1834); and (in con- 
nexion with Hamilton’s principal function) by Sir W. R. Hamilton, “On a 
General Method &c.” (1834), p. 282; Jacobi, “Zur Theorie &e.” (1837), .' 
p- 122; Cayley, ‘ Note on the Theory of Elliptic Motion” (1856). 
22: Connected with the problem of central forces, we have Sir W. R. 
Hamilton’s ‘ Hodograph,’ which in the paper (Proc, R. Irish Acad, 1847) is 
defined, and the fundamental properties stated; viz. if in an orbit round a 
eentre of force there be taken on the perpendicular from the centre on the 
tangent at each point, a length equal to the velocity at that point of the orbit, 
the extremities of these lengths will trace out a curve which is the hodograph. 
As the product of the velocity into the perpendicular on the tangent is equal 
to twice the area swept out in a unit of time (vp=h), the hodograph is the 
reciprocal polar of the orbit with respect to a circle described about the centre 
of force, radius =/h. Whence also the tangent at any point of the hodo- 
graph is perpendicular to the radius vector through the corresponding point 
of the orbit, and the product of the perpendicular on the tangent into the 
corresponding radius vector is =h. | 
wn 
If force q& (dist.)—2, the hodograph, qua reciprocal polar of a conic section 
with respect to a circle described about the focus, is a circle. 4 
23. The following theorem is also given without demonstration ; viz.if two — 
circular hodographs, which have a common chord passing or tending through 
a common centre of force, be both cut at right angles by a-third circle, the 
times of hodographically describing the intercepted arcs (that is, the times of — 
describing the corresponding elliptic ares) will be equal. 
24, Droop, “‘On the Isochronism é&e.” (1856), shows geometrically that . 
the last-mentioned property is equivalent to Lambert’s theorem; and an 
analytical demonstration is also given, Cayley, ‘A demonstration of Sir W. 
‘R. Hamilton’s Theorem &e.’’ (1857). See also Sir W. R. Hamilton’s ‘ Lee- 
tures on Quaternions’ (1853), p. 614, 
25. The laws of central force which have been thus far referred ~ are force 
ar, os: Cie ; and it has been seen that the case of a force P+ depends 
