40 
Proceedings of Royal Society of Edinburgh. [jan. 7 , 
by (1) and (6), the complete solution of the problem : for g and F 
any given arbitrary functions. 
It is obvious that, so far as is concerned, the general solution 
for x any considerable multiple of ± l , and exceeding ± l by not 
less than two or three times the wave-length, 27ry/co 2 , must, for 
values of t great enough to have let the front of the procession pass 
the place x, be 
sin 
+ A cos r o)t - — ( - f)"l 
L 9 J 
L 9 J 
and 
for x positive, 
= sin |^a)£ - ( - x +/) - A cos j^c ot - - x + f)J 
for x negative, 
H10); 
where 51 and/ denote quantities calculable from the form of § ; 
and A and f similarly from F. Further, it is obvious that the 
front of each procession will, for any value of t not less than 
several times the period and not less than several times the time one 
of the wave-crests takes to travel through a space equal to l, be 
independent of the particular forms of § and F. From the theory 
of Stokes, Osborne Reynolds, and Rayleigh, we know that it 
advances at half the speed of a wave-crest; but their theory, so 
far as hitherto developed, does not teach us the law according to 
which the front, as it advances, becomes longer and longer in pro- 
portion to J t , nor even the fact that it does become longer and 
longer. All the details of this interesting question are exquisitely 
given in what follows : having been found with great ease for the 
particular case, 
IX*) = 0, and g(a) = j } * . . . (11), 
where b denotes a length of any magnitude, which we shall take to 
be very small in comparison with 27 ry/w 2 , the wave-length, We 
shall in fact find that 
iVo)- C + 
{ 
{xZ+b^ + b } * 
x i 
+ 6 2 
1 
sin mi 
( 12 ), 
in the particular processional case of the general equations 
(1) . . . (6), which we now go on to work out. 
