the Isochronism of the Circular Hodograph. 429 
the centre of the orthotomic circle (where R is arbitrary) ; the 
equation of the polar of this point with respect to the directrix 
circle 7*=ma cos x, is then at once seen to be 
rcos (@—a) = ae 
which is the equation of the line cutting off the are of the elliptic 
orbit 
gu} a cos? K 
~ 1+sin«sin (@—a)’ 
Writing 0O-—w=0—2+(a—a), the two equations give 
A 
cos (0—«) = Py 
sin (@—a)= 20) 
if for shortness 
MA COS Kk 
ee sae 
Ba macs e cos(a—@) =a cos?@x 
~  R sm(¢—se) sine sin (e¢—a)’ 
Ni 
rs 
sin « sin («—a) ” 
we have therefore 
A?+B? 2BC 
ata + eae +(C?= ¥; 
or, what is the same thing, 
(1 —C?)r?—2BCr —(A?+ B’)=0; 
and thence, if 7’, 7’ are the two values of 7, 
2BC 
gf 47! ic? 
A? + B? 
Three 
a 1c?’ 
Let 6', 6" be the corresponding values of 6, we have 
6'—6"=0'—a—(6"—2), 
and thence 
Bo te B ae 
cos (0-0) = 5+ (540) (a +0) 
A? + B? 
hw t 
= ill +BO (33+ ai) +6 
