480 Or the Isochronism of the Cireular Hodograph. 
Or adding unity to each side, multiplying-by 7’ 7", and on the 
right-hand side substituting for 7! +7", 7/r" their values 
3 2A2 
(1+ om 0) = 228, 
the square of the chord is 72+ r!2—2rv! cos (6' — 6"), or, what is the 
same thing, (7! +7")? 27’! @! + cos (6'—8")) ; hence to prove 
the theorem, it is only necessary to show that 7/+r" and 
"71 + cos (6'—6")) have the same values in each orbit, that 
2BC 207A2 
is, that i-@ and — 1T=C? 
But observing that 
have the same values in each orbit. 
1— sin? « sin? («—@) = cos? «+ sin? « cos’ (a—@) 
= cos*«{1+ tan*« cos* («—@) }, 
the values of these expressions are respectively 
oa (m tan x cos («—@) —R) 3 
R-~ 1+ tan? « cos? (ea—a) 
RK? 1+ tan? « cos? (2—a@)’ 
which contain only the quantities m, a, R, tan « cos (a —@), which 
are the same for each orbit, and the theorem is therefore proved, 
viz. it is made to depend on Lambert’s theorem. I may remark, 
that a geometrical demonstration which does not assume Lam- 
bert’s theorem is given by Mr. Droop in his paper “ On the Iso- 
chronism of the Circular Hodograph,” Quarterly Math. Journal, 
vol. i. pp. 374-3878, where the dependence of the theorem on 
Lambert’s theorem is also shown. 
By what precedes, the theorem may be stated in a geometrical 
form as follows :—“ Imagine two ellipses having a common focus, 
and their major axes equal; describe about the focus two direc- 
trix circles having their radii proportional to the square roots of 
the minor axes of the ellipses respectively ; the polar reciprocal 
of each ellipse in respect to its own directrix circle will be a circle 
(the hodograph), and the common chord or radical axis of the 
two hodographs will pass through the focus. Consider any point 
on the common chord, and take the polar with respect to each 
directrix circle ; such polar will cut off an are of the correspond- 
ing ellipse; and then theorem; the elliptic chord, and the sum of 
the radius yectors through the two extremities of the chord, will 
be respectively the same for each ellipse.” 
2 Stone Buildings, W.C. 
June 24, 1857. 
