1903-4.] Lord Kelvin on a Free Procession of Waves. 
311 
On the Front and Rear of a Free Procession of Waves 
in Deep Water. ( Continued from Proc. R.S.E., Feb. 1st, 
1904.) By Lord Kelvin. 
(Read June 20, 1904.) 
§ 11. The present communication is substituted for another 
bearing the same title, which was read before the Royal Society 
of Edinburgh on January 7th, 1887, because the result of that 
paper was rendered imperfect and unsatisfactory by omission of 
the exponential factor referred to in § 10 of my paper of February 
1st, 1904. I shall refer henceforth to the last-mentioned paper as 
§§- 1 . . . . 10 above. 
§ 12. I begin by considering processions produced by super- 
position of static initiating disturbances, of the type expressed in 
(12) of § 4 above; graphically represented by fig. 1 ; and leading 
to. motion investigated in §§ 1-3, 5-10. The particular type of 
that solution which I now choose, is that chosen at the end of § 4, 
which we, with a slight but useful modification,* may now write 
as follows — - 
where p = J(z 2 + x 2 ), and X= tan 1 (x/z) 
Here - £ denotes the upward vertical component of the displace- 
ment of the fluid at time t from its undisturbed position at point 
(x, z), which may be either in the free surface or anywhere below 
it. Taking £ = 0 in (17), we have, for the initial height of the 
free surface above the undisturbed level, 
(18). 
§13. We shall first take, as initiating disturbance, a row 
oo to +oo of superpositions of (18); alternately 
extending from 
* The substitution of bX, for iir-tan -1 . / LL? saves considerable labour 
' v P-F 
and use of logarithms ; especially when, as in our calculations, z = 1. 
