1915 - 16 .] On Systems of Partial Differential Equations, Etc. 311 
for the requirements are satisfied by putting x p = x, yp — y, z v = z, t P = t 
(p = l, 2, . . . n). 
Let us see if this is the only solution. We easily find that 
0W_^0Va% ^8V d\, 
8x 2 _ “5^ ox 2 + 2^dy p dx 2 + ^ dz p dx 2 
A " 3W dxp dxg 
+ 2-i 2-i dx v dx a dx dx + ’ ' * 
p= 1 2=1 P 1 
y,3V 0% 3Y 3% 
a«-2 + Za 0/ g^2 
P = 1 P 
^ aw + 
"1" 2—i j dx p dy q dx dx 
P— 1 q — 1 r * 
If the equation of wave-motion is to be satisfied when V is an arbitrary 
multiple wave-function, all the sets of equations of type 
dx p 8 y q dx p dy q dx v d y q 
1 dx p dy q 
'x a t a t 
dx dx 9 y dy dz d z 
must be satisfied (p, q — 1, 2, . . . n), and these seem to imply that the 
variables x q , y q , z q , t q differ from constant multiples of the variables x p , y p , z p , t p 
by arbitrary constants. This solution of the problem is not of much 
interest. If, however, we limit V to be a completely neutral multiple wave- 
function, that is, a function which satisfies all the equations of type 
a 2 v aw 
in addition to the equation 
a 2 v 
dx p dy q dx q d y p 
a 2 Y 8 2 V 
dx p dx q ' dy p dy q ' dz p dz q 
1 a 2 v 
C 2 dt p dt q 
then the conditions to be satisfied by the variables x p , y p , z p , t p are not so 
numerous. We have, for instance, to satisfy equations of type 
dx v dy n dx v dy q dx p dy q 1 dx p dy q dx q dy p dx q dy p dx q dy p 1 dx q dy p _ 
dx dx dy dy ^ dz dz c 2 dt dt ^ dx dx + dy dy dz dz c 2 dt dt 
Adopting linear expressions in x, y, z, and t for each of the variables 
x p , y P , z p , t p , we are led to the following expressions: — 
x p = y p x - h p y + g p z + if v ct + £ p 
y p = lipX + {Xpij - fpz + ig p ct + rj p 
z P = - g P x + f p y + y p z + ihpd + 
ct , 
( 11 ) 
»p _ if v x + ig P y + ihpZ + y P ct + r p 
where y p , f p , g v , h p , £ p , rj p , £ p , t p are arbitrary constants. With these ex- 
pressions for x p , y p , z p , t p the wave- equation 
aw aw aw_iaw 
dx 2 + dfi + ^~"ddt^ 
is satisfied whenever Y is a competely neutral multiple wave-function. 
