209] 
OF THE ISOCHRONISM OF THE CIRCULAR HODOGRAPH. 
265 
Let 6', 6" be the corresponding values of 6, we have 
e'-d" = 0'-a-(d"-a), 
and thence 
co s«r —*")=£.£, + (*+c) 
~J+bc(\ 
r r \r 
or adding unity to each side, multiplying by r'r", and on the right-hand side sub 
stituting for r + r", r'r" their values 
/V(l + co S (0'-<T)) = -^; 
the square of the chord is r 1 + r'- — 2rr cos (6' — 6"), or, what is the same thing, 
(?■' + r"f — 2r'r" (l + cos (O' — 0")) ; hence to prove the theorem, it is only necessary to show 
that r' + r" and r'r" (l + cos (6' — 6")) have the same values in each orbit, that is, that 
2BG , 2C 2 M 2 . . . , ! • -o ! • , 
1~(R ancl have the same values m each orbit, lout observing that 
1 — sin 2 k sin 2 (a — zx) = cos 2 k + sin 2 k cos 2 (a — zj) = cos 2 k {1 + tan 2 k cos 2 (a — ta-)}, 
the values of these expressions are respectively 
2a (m tan k cos (a — ot) — R) 
R 1 + tan 2 k cos 2 (a — ot) 
2a 2 to 2 
R- 1 + tan 2 k cos 2 (a — ot) ’ 
which contain only the quantities m, a, R, tan k cos (a — to), 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 
Lambert’s theorem is given by Mr Droop in his paper “On the Isochronism of the 
Circular Hodograph,” Quarterly Mathematical Journal, vol. I. [1857] pp. 374—378, 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 directrix 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 arc of the corresponding ellipse; and then, theorem, 
the elliptic chord, and the sum of the radius vectors through the two extremities of 
the chord, will be respectively the same for each ellipse.” 
2, Stone Buildings, W.C., June 24, 1857. 
C. III. 
34