1905-6.] Lord Kelvin on an Initiational Form. 
399 
Initiation of Deep-Sea Waves of Three Classes : (1) from 
a Single Displacement ; (2) from a Group of Equal 
and Similar Displacements ; (3) by a Periodically 
Varying Surface-Pressure. By Lord Kelvin. 
(Read January 22, 1906. MS. received October 15, 1906.) 
(1) Disturbance due to an Initiational Form more convenient 
THAN THAT OF §§ 3-31 OF PREVIOUS PAPERS ON WAVES. 
§§ 96-113. 
§ 96. The investigations of §§ 5-31, including the “ front and 
rear ” of infinitely long free processions of waves in deep water, 
are all founded on initiational disturbances, according to the 
first of two typical forms described in §§ 3, 4. In this form 
the initial disturbance is everywhere elevation or everywhere 
depression, and its amount, at great distances from the origin 
varies inversely as the square root of the distance p, from a 
horizontal line at a small height h above the water-surface in the 
middle of the disturbance. In the present paper a new form of 
type-disturbance is derived indifferently from either the first or 
the second, of the forms of §§ 3, 4 : from the first, by double 
differentiation wfith reference to time, t ; from the second, by 
single differentiation with reference to space, x. 
§ 97 (being a repetition of §§ 1, 2, slightly modified with respect 
to notation). Consider a frictionless incompressible liquid, (called 
water for brevity,) in a straight canal, infinitely long and infinitely 
deep, with vertical sides. Let it be disturbed from its level by 
any change of pressure on the surface, uniform in every line 
perpendicular to the plane sides, and let it be left to itself under 
constant air pressure. It is required to find the displacement 
and velocity of every particle of water at any future time. Our 
initial condition will be fully specified by a given normal 
component of velocity, and a given normal component of dis- 
placement, at every point of the surface. 
Taking 0 , any point at a distance h above the undisturbed 
water level, draw 0 X parallel to the length of the canal, and 0 Z 
vertically downwards. Let £, £ be the displacement components, 
