If « is even. 
If n is odd, 
of the Crossing of Rays. 
\/'2 — 2 cos oirp . cos 2ir(t— jY 
x=v-2 + 2 cos 37T£> . sin 2-7T 1 1 — ^-Y . 
61 
(5) 
. . (6) 
Equation (4) shows that along these lines the vibrations are 
rectilinear, and perpendicular to direction of the ray. 
Putting p = -7T, m being an integer, we see from (5) and 
i 
(6) that there are points of no motion when m and n are both 
even or both odd. These conditions will be satisfied if p, q, 
and r are multiples of -t. In order to satisfy (1), one of the 
quantities must be an even multiple, and the other two must 
be both even or both odd. 
TVe may obtain similar equations in relation to Q'OQ and 
R'OR; and the points of no motion will be the same as those 
already obtained. If we draw the three sets of lines above 
considered, we shall form a series of triangles whose sides are 
parallel to the rays, each side being equal to f . These tri- 
angles will have the properties, that their angles will be nodes, 
and that the vibrations along their sides will be perpendicular 
to the sides, the displacement being given by equations (5) 
and (6) and the corresponding equations for the rays Q and R. 
The form of these triangles under displacement, when r=0, 
is shown in PI. I. fig. 1. 
The motion may be also readily obtained along lines per- 
pendicular to the direction of the rays, at distances ^ from 
each other, one of each set passing through the origin, p must 
be a multiple of ^; and there are six different forms of equa- 
tions (2) and (3) for six consecutive values of p, which are 
given in the following Table [n being an integer): — 
p- 
X. 
y- 
2ft + 1 
(1+A)sm27rf 
— B cos 27r£ 
2n + I 
(l-A)sin27r(Y + i) 
+ Bcos2*r(* + ^) 
2n+l 
(1 + A)sin27ra-1) 
— Bcos2ir(t— £) 
2n 
(1 — A)sin27rf 
+ B cos 2-Trf 
2n-l 
(1 + A)sin27r(£ + 1) 
-B cos 2ir(t + 1) 
2«-| 
(l-A)sin27r(^-i) 
+ B cos 2?r(£ -A) 
where A= cos7r(g — r), B= </2> sin7r(^ — ?■). 
