410 NEUMANN ON A METHOD OF ESTIMATING THE 
d?u- 3d?u Pu 2d? vu >) 
ee Naar tay dudy 
@u_dv _3dv , 2d*u (1.) 
di dae * dy? 
We can remove the third plane of vibration moving perpendi- 
cularly to its plane, from the two first equations, by introducing 
the condition— 
Se) ae erate Se col Se 
whereby they become changed to— 
MG du dP a 
Dae dat? ay 
Lay eo ao 
ad de®” d¥ 
A special integral of the equations (3.) which satisfies the condi- 
tion (2.), exists when the plane of vibration forms the angle ¢ 
with the axis y: 
(3.) 
SS fe — Asin g sin (228¢ + 98m? _ 7), 
A 
. (4) 
v= Acos ¢sin (79S + ¥5Ne _ = 2, 
in which a T? = A, A expressing the length of the undulations, 
and T their duration. From the third equation in (1.) we obtain 
. (xcoso¢+ysingd ¢ 
~ = C sin ee <, we . (5.) 
These special integrals are sufficient to allow of the deduction of 
the laws of reflexion and refraction. Thus, if we suppose the 
medium to border closely upon a second in the plane of y and z, 
a reflected and transmitted vibration exists at this border. The 
reflected vibration must require the same differential equations 
as the incident, it is also parallel with the axis z, but forms the 
angle 180 — ¢ with y, hence, when its vibrations are expressed 
by w’, v’, and w’, 
be ee . = Sees Tee) 
wu’ = A!sin 9 sin (=sees tysne T 
Bene Bie Gs ea re = ree ok lea 
parece or MP 8 *) on | 
/ View 
vw = C'sin ( 
A 
ee 
