14 Davies.—Some Electric and Elastic Analogies. 
■will constitute a simple simultaneous solution of the equations. On 
substitution of such a set of values the equations take the form 
( < 5 2 ^ 8 i v i ) d^r i 
m \ r <T&+ r i'm*s r i‘mh) +nr y‘ u i—p u Cfflr 
with similar equations for v and w each. 
The necessary and sufficient conditions for a solution are that 
T i —r^z=zJ' for all values of i. Because on this supposition left hand 
members can be freed from time functions and the right hand members 
from position functions; in other words, we have a separation of the 
variables, and the equations may be written 
m 
8 { 
; s Ui 
•+- 
6 Vi 
9 
i 
' 8x ( 
1 8x 
Sy 
r i 
dP 
m 
8 ' 
: $u t 
SVi 
swf ; 
P 
V 
\$y | 
/ 8x 
“h 
8y 
* 6z 
r i ‘ 
‘ dP 
m 
8 ( 
i *«( 
1 . 
8Vi 
, Sw i | 
> n 0 
_ _ rv ^ a i*i —— 
p 
d i r i 
~ w ~i 
’ 8z ( 
f 8x 
“T 
8y 
+ 8z 1 
* Ar [7 tv * — 
) w i 
T i 
' df 
In these equations three functions independent of time are equated to 
the same function of time and not position. They must therefore all 
be constants. Let that constant be denoted by ip (negative) so that 
also 
p d' 2 ri 
m 
m 
V 
dP 
8 \ 
i Sw i 
8v, 
i i 
Sw i\ 
8x | 
1 8x 
' Sy + 
6z f 
' Su. 
8V; 
Sw i \ 
h l 
1 8x 
Sy + 
Sz Ij 
8 < 
; Su i 
8v • 
i 1 i 
S_Wj ) 
1 8z | 
( Sx 
Sy + 
8z jj 
=—ip 
+ np r *v i + piv i =0 
with solutions u—'S{u i r^ 
v=S(v i T i ); W=2(w ( r { y, 
