428 Mr. A. Cayley on Sir W. R. Hamilton’s Theorem of 
determine the time of description of the are; and the values of 
these quantities, viz. the elliptic chord and the sum of the radius 
vectors, must be the same in each orbit. 
The analytical investigation is not difficult. I take as the 
equation of the first orbit, 
pelt Coe i ere 
"= T+ ecos (0@—a)’ 
then the polar of the orbit with respect to a directrix circle r=c is 
Ache cas MAGES Rey ON SU a 
r meee @) mae) =0. 
And putting =k Wk Va(1—e) (where k& is a constant quan- 
tity, 2. e. it is the same in each orbit), the equation becomes 
kvk 
iat kV ke e pcos (@—a)— Ke =0. 
V a(1—e?) a 
But since a is supposed to be the same in each orbit, we may 
for greater simplicity write 4®=m?a; it will be convenient also 
to put e=sin«; we have then 
ast a cos* K 
~ 1+sin «cos (@—a@) 
for the equation of the orbit, 7?=ma cos « for the equation of the 
directrix circle, and 
r 
r?—m tan xr cos (@—w)—m?=0 
for the equation of the hodograph. 
We have in like manner 
i a cos? x! 
"= T+ sink cos 0—a) 
for the equation of the second orbit, 7?=ma cos x! for the equa- 
tion of the corresponding directrix circle, and 
r?—m tan x’r cos (@—a') —m?=0 
for that of the hodograph. 
The equations of the two hodographs give at once 
tan « cos (9—a@) — tan x! cos (9—a') =0 
for the equation of the common chord or radical axis of the two 
hodographs,—an equation which shows that, as already noticed, 
the common chord passes through the origin or centre of force. 
This equation gives @=a if 
tan x cos («—a@) — tan x! cos (a—a') =0; 
i, e. ais a quantity such that the expressions tan x cos e—@ and 
tan x! cos(«—a!), which correspond to each other in the two 
orbits, are equal, We may take R, « as the polar coordinates of 
