132 
where s = S/a and r = b/a\ or, what is the same thing, 
(16) 
i + r ’ 2 
1 — r 2 
5 2 
— sec 2 ty . 
Writing now u for the ratio a 2 /a 1 ( or f° r the inverse ratio a 1 /a 2 ) 
we conclude from (6) that 
(17) u* — cos 6 + 1 = 0. 
Making use of (11) and (16): 
(18) u^ + 2u 1 ^-,-\-l = 0. 
1 — s 2 ' 
If we confine ourselves to values of u and s included between 0 
and 1 we can write 
(19) 
u 
1 — s 
1 + s ' 
§ 2. In order to test the results so far obtained let us go back 
to (3), § 1. Suppose a new system of coordinate axes x' : y' be in¬ 
troduced, inclined at angle W to the original axes x , y\ and let 
(1) 
I' = A cos n(t — y) 
r\' = B cos n (t — v) 
Then it is easily verified that 
(2a) A 2 = a 2 cos 2 ¥ -|- b 2 sin 2 ¥ -[- 2 a b sin ¥ cos ¥ cos I) 
(2b) B 2 = a 2 sin 2 ¥ -(- b 2 cos 2 ¥ — 2 ab sin ¥ cos ¥ cos D . 
If we assume as a particular case ¥ = ip we have 
(3) 
A = a and B — é . 
With the help of (6), (8) and (11) in § 1. we thus obtain 
(4) a 2 = (a! + a 2 ) 2 ; = (% + a 2 )' 2 
and this is in conformity with our previous result. 
If we take a x =a 2: the motion will be a simply periodic linear 
oscillation; it becomes circular when 
(Ö) 
= 0 or a 2 — 0 . 
