424 Proceedings of Royal Society of Edinburgh, [ sess . 1905-6.] 
In fig. 39, the inscriptions of times correspond to cos w(t - q) in 
the formulas. The same curves, with the inscriptions altered to 
(i + 2/8)r, (i + 3/8)r, (i+ 4/8)r, (^ + 5/8)r, (i + 6/ 8)r, correspond to 
sin <jo(t - q) in the formulas. 
§ 139. In fig. 36, representing velocity-potentials and a surface 
displacement, none of the curves shows any perceptible deviation 
from sinusoidality except within period 1. Towards the end of 
period 1 the numbers found by the quadratures show deviations 
from sinusoidality diminishing to about 1/10 per cent., and imper- 
ceptible in the drawings. This proves that sinusoidality is exact 
within 1/10 per cent, through all time after the end of the first 
period. 
It is interesting to see, in period 1, how nearly the rise from 
the initial zero follows the same law for S^(0 , 1 , t) and 
S^,(0 , 1 , t) : notwithstanding the vast difference in the law of 
initiating surface-pressure, represented by (188), for these two 
cases. In fig. 36, the initiating surface-pressure commences 
suddenly at its negative maximum value, — J 2 for case </> , and 
-•5 for case if/, of which the former is 2-83 times the 
latter. The semi-amplitudes of the subsequent variations of 
velocity-potential shown in the first and third curves are - 954 
for case </> and ‘318 for case if/, of which the former is 3'00 
times the latter. 
§ 140. The first, and third, and fifth, curves of fig. 37 show, at 
a distance of one wave-length from the origin, the complete history 
of velocity-potential and of surface displacement through all time 
from the beginning of application of pressure to the surface. The 
very approximately accurate sinusoidality of each of these three 
curves through periods 6, 7, 8, shows that the continuation through 
endless time is in each case sinusoidal. 
In remarkable contrast with the initial agreement between 
S^(0 , 1 , t) and S^(0 , 1 , t) , to which we alluded in § 139, we find 
very instructively a remarkable contrast between S^(8 , 1 , t) and 
S<^(8 , 1 , 7)- throughout the whole of the first period. Remembering 
that in a liquid of unit density the pressure is equal to minus the rate 
of augmentation of the velocity-potential per unit of time, and remark- 
ing that the displacement £^(0 , 1 , t) is, as is shown in its curve, 
very nearly zero throughout the first period, and that £^,(0 , 1 , t) is 
