416 Proceedings of Royal Society of Edinburgh. [sess. 
Reynolds’ * important contribution to the ideal doctrine' of 
“group-velocity.” 
§ 126. The dynamical conclusion of § 125 is very important and 
interesting in the theory of two-dimensional ship-waves. It shows 
that the approximately regular periodic train of waves in the rear 
of a travelling forcive, investigated in §§ 48-54 and 65-79 above, 
cannot be as much as half the space travelled by the forcive, from 
the commencement of its motion ; but that it would be exactly 
that half-space if some modifying pressure were so applied to 
the water-surface in the rear as to cause the waves to remain 
uniformly periodic to the end of the train ; without, on the whole, 
either doing work on them, or taking work from them. 
A corresponding statement is applicable to our present subject, 
as we shall see in §§ 156, 157 below. 
§ 127. Go back to § 118; and first, instead of a sinusoidally 
varying pressure, imagine applied a series of impulsive pressures, 
each of which superimposes a certain velocity-potential upon 
that due to all the previous impulses; and let it be required to 
find the resulting velocity-potential at any time t, after some, 
or after all, of the imjDulses. Consider first a single impulse at 
time t -q ; that is to say, at a time preceding the time t by an 
interval g. Let the velocity-potential at time t, due to that 
single impulse applied at the earlier time t -q, be denoted by 
CV(jj, z, q) (168). 
According to this notation the instantaneously generated velocity- 
potential is CV(£, z, 0), and the value of this at the bounding 
surface of the water is CV(x, 1, 0). Hence, by elementary hydro- 
kinetics, if I denotes the impulsive surface-pressure, we have 
1= -C Y(x, 1, 0) (169). 
§128. Considering now successive impulses at times preceding 
the time t , by amounts q Y , q 2 , .... q^, and denoting by 
S(x,z,t) the sum of the resulting velocity-potentials at time t , 
we find 
S(a;, a, t) = Oft(x, z. , q Y ) + C 2 V(:r, z, q 2 ) + .... . CiY(^, z, ft) (170). 
Supposing now the impulses to be at infinitely short intervals of 
* Nature , August 1877, and Brit. Ass. Report , 1877. 
