100 Proceedings of the Royal Society of Edinburgh. [Sess. 
h_ 
fh , \ 
h {h ^ \ 
2 
+ 00 
"2“l2 + ’'d 
27T _ 
^ C ~ - ^o)‘^ + V ^j(p^ - 4)^ + (^ + Vo)^ 
-00 
+00 
-00 
= Z 
/ - 
Ho 
+ 00 
-GO 
+ Z 
2M^qP 
+ ^ h2_p2l2 
(7^2 + pH^f 
^ // 1 V 1 
+ ^ 2^2 + _^(;,2 + ^ 2 ^ 2 ;> 2 j ’ 
( 8 ) 
on neglecting terms of the second order in and and equating 
P 
to zero. In the same way 
- —V. 
7T 1^, 
= Z' 
- 2 , 
Pl-io 
+ CO 
2 
00 
r+oo 
1+A 
qrqV\ 
^ r t ^ ~ 2az»7jp n 
~ “ ^"(FTioW “ (F++W J 
L/j2 + J)2/2 ’«(7i2 H- y2;2)2 (/j2 ^ y2;2)2 
= fo 
/I -pH-i 
( 9 ) 
If the whole fluid be given a velocity ^ 772 ^2 parallel to X in the 
negative direction, we may omit that term in equation (8), and substituting 
di. 
d 
and for and respectively, these equations become 
dt 
dt 
where 
and 
0 
s 
1 
11 
5 S; 1 "^ 
. 
■ ( 10 ) 
^ Ao = aA . . , 
( dt 
• (11) 
' 1 
W"" -t- p’-'>f 
I — 1 
d% o?i\ 
* 2 “ 4 ^ 2 fo- 
, . (13) 
dt^ ^ 4+^" ■ 
• (14) 
Since a cannot vanish,^ these equations imply that no matter what 
relations exist between h and I, and tend to increase exponentially 
* From equations (66) and (70) in Appendix it is easily found that a= — ^ 
Z2 sinh2 
