1905-6.] Lord Kelvin on the Growth of a Train of Waves, 435 
of E, in (205), and (206), Jmt is very great in comparison with 
ayfijm, the two added terms in (205) are approximately equal, 
and (206) is reduced approximately to its last term ; and all the solu- 
tions (189), (190), become approximately sinusoidal, in respect to t. 
This is the case when t is very great in comparison with 
unity, and in comparison with <o V 5 as we see by looking at 
the moduluses shown in (209), (210), (211). This allows us to 
neglect w in the arguments of E in (205), (206), and makes P 
and Q constant relatively to t. 
§ 157. When t is small or large, and x not so small as to give 
preponderance to the first terms of the moduluses (209), (210), we 
have in (205), (206), (189), (190) a full representation of the 
whole circumstances of the wave-front, extending from oc back 
9 
to the largest value of x that allows preponderance of 
V -, in the moduluses, (209), (210). Let, for example, 
ix 
t. /JL = w ,x 
ix 
( 212 ). 
This gives 
x — = half the wave- velocity . . . (213). 
2oj 
The moving point thus defined is what in my first paper to the Royal 
Society of Edinburgh (January 1887), “On the Front and Rear of a 
Free Procession of Waves in Deep Water,” I called the “ Mid-Eront,” 
defined in (45) of that paper, which agrees with our present (213). 
The following passage was the conclusion of that old paper : — 
“ The rear of a wholly free procession of waves may be quite 
“ readily studied after the constitution of the front has been fully 
“ investigated, by superimposing an annulling surface-pressure upon 
“ the originating pressure represented by (12) above [this is a case 
“ of (173) of our present paper], after the originating pressure has 
“ been continued so long as to produce a procession of any desired 
“ number of waves.” The instruction thus given with reference to 
the relation between front and rear has been virtually followed, 
though with some differences of detail, in §§ 20-24 of my second 
