131 
we have 
a 2 — a x 2 - 1- a 2 2 -|- 2 a x a 2 cos 6 (6a) 
Let 
b 2 = a y 2 -j- a 2 — 2a x a 2 cos Q . 
(6b) 
D = n (a — ß) ; 
CO 
by a well-known proposition, the angle ip which the principal axes 
of the ellipse make with the coordinate axes x, y is given by the 
equation 
2 ab cos D 
tg 2 = — -—— 
° a 2 — b 2 
( 8 ) 
Now from (7) and (4) we obtain 
a b cos D = 2a x a 2 sin Q 
and from (6): 
a 2 — b 2 — 4 a x a 2 cos Q 
Combining (8), (9), (10) we find 
(9) 
( 10 ) 
tg 2 if) = tg 8 . (11) 
The same results, evidently, would have been obtained if we 
had conceived the vibration represented by (3) as the resultant of 
the two following 
(12a) 
(12b) 
If a and C are the semi-axes of the ellipse, it is easy to show 
that the following relations will hold: 
a 2 + 6 2 
<Z 2 + £ . ^ 
— sin 2 b 
(13) 
a 2 b' 2 
aH* 
and 
a 2 -\-b 2 
— a 2 -j- 6 2 . 
(14) 
Eliminating D between (13), (14) and (8) we obtain 
s 2 (1 + r*y 
(jf — r 2 ) 2 tg 2 2ip 
(16) 
r 2 (1 + s 2 ) 2 
r 4 r 2 
