ON THE SPECIAL PROBLEMS OF DYNAMICS. 189 
dt= il RT 
{hr (C—2f Pdr)}* 
ja hdr 
r{—h-+7°(C—2f Par)}" 
which, however, assume that P is a function of r only. 
15. Force & (dist.)-2._ The law of motion in the conic sections is implicitly 
given by Newton’s theorem for the equable description of the areas, For the 
parabola, if « denote the pericentric distance, and f the angle from pericentre 
or true anomaly, we have 
pace Me (tan 2f+ 3 tan® if). 
Nv 
For the ellipse we have an angle g, the mean anomaly varying directly as 
the time (g=nt if nave ; an auxiliary angle u, the excentric anomaly, 
az 
connected with g by the equation 
g=u—esin u; 
_and then the radius vector r and the true anomaly f are given in terms of w 
by the equations r=a (1—e cos w), and 
e086 gin pM Ie sin and «tan afm 1 tan de, 
—e 
co —— —— —— = 
f l—ecosw 1l—e cos u 
16. It is very convenient to have a notation for and f considered as func- 
tions of ¢,g, and I have elsewhere proposed to write 
r=a elqr(e, 9), f=elta (e, 9), 
read elqr elliptic quotient radius, and elta elliptic true anomaly. 
17. The formule for the hyperbola correspond to those for the ellipse, but 
they contain exponential in the place of circular functions (see post, Elliptic 
Motion). 
18, Euler, in the memoir “Determinatio Orbitz Comete Anni 1742,” 
(1743), p. 16 et seq., obtained an expression for the time of describing a para- 
bolic are in terms of the radius vectors and the chord; viz. these being f, g, 
and k, the expression is 
Time arrAl (r+94%)'— (r+0-z)'T, 
which, however, as remarked by Lagrange, ‘ Méc. Anal.’ t. xi. (3rd edit. p. 28), 
is deducible from Lemma X, of the third book of the ‘Principia.’ But the 
theorem in its actual form is due to Euler. 
19. Lambert, in the ‘ Proprietates Insigniores, &e.’ (1761), Theorem VII, 
Cor. 2, obtained the same theorem, and in section 4 he obtained the corre- 
sponding theorem for elliptic motion ; viz. the expression for the time is 
at —¢'—(sin ¢—sin » 
Paige 9—o g $ 
