60 Mr. W. Baily's Illustration 
tions to straight linos perpendicular to P' OP, Q' Q, R' R 
respectively; and the equations q — r = const, r—p — const., 
p—q = const, are equations to lines parallel to P'OP, Q'OQ, 
R'OR respectively. When the constant is zero, the lines pass 
through 0. 
If we take any point in Q'Q and move perpendicularly to 
Q' Q from this point, we can, without altering the phase of 
the vibration of the ray Q, reach a point at which the 
phase of the vibration of the ray R is the same. If we 
now move from this latter point in a direction parallel to 
P' P, we shall keep the phases of Q, R equal to one another, 
and we can reach a point at which the phase of the ray P is 
equal to either of them. Take this point as the origin, and let 
the phases be zero at the initial time. Then at a time t the 
displacements due to the three rays at the point T will be 
sin 27r(t— p), sin 2*7r (£—</), sin '2ir (t — r) , the wave-length 
being taken as the unit of length, and the period as the unit 
of time. 
Let x be the amount of displacement along TX and y that 
along TY, at the time t. Then 
x — sin ^ sin 2ir{t —p) + sin ( - -j- -^- ) sin 2ir(t — q) 
(ir 2ir\ . . v 
2 "~ -o-)sin27rU— r), 
y = cos j> sin 2ir(t —p) + cos ( ^ + -^- ) sin 2ir(t — q) 
/it 27r\ . _ . . 
+ cosl-^ ^- Jsin ziryt— r). 
By means of (1) these equations may be written 
x=sm2'n(t— ])) — cos ir(q — r)s'm2ir(t + J 7 ), . (2) 
y= s/ S sin ir(q — r) cos 2-7T (^ + ^ ) (3) 
In general the calculation of the phase and the ellipse would 
be laborious; but it may be readily effected along lines parallel 
to P'OP, Q'OQ, R'OR at distances — — j from one another as 
s/ 6 
follows : — We have as equation to such lines parallel to P'OP, 
q—r=n, where n is an integer. Hence 
y=o, W 
x= sin 27r (t—p) — sin 27rf / + { J cos nir. 
