406 Proceedings of Royal Society of Edinburgh. [sess. 
surface displacements according to our new solution if/(x , z, t) for 
the six values of t respectively, 0 , J^/tt, n /it, ^ Jit , 8 ^/ 7 r. 
The formulas are simplified by taking g = 4. This is merely 
equivalent to taking as our unit of length half the space descended 
in one second of time, by a body falling from rest under the 
influence of gravity. For simplification in the writing of formulas 
we take z = 1 for the undisturbed level of the water-surface. The 
subsidiary curves, explained in § 107 below, are called argument- 
curves, as they represent the argument of the cosine in (144). 
§ 106. One exceedingly curious and very interesting feature of 
these curves is the increasing number of values of x for which the 
displacement is zero as time advances, and the large figures, sixteen 
and sixty-four, which it reaches at the times, ijir and 8 ^/ 73 -, of the 
last two diagrams. These zeros, for any value of t, are given by 
the equation 
A = (2;+1)tt/2 (146). 
§ 107. Notwithstanding the highly complicated character of the 
function represented in (145), the zeros are easily found by tracing 
an argument-curve, with A as ordinate, and x abscissa (as shown 
on the ir-positive halves of the six diagrams on two different scales 
chosen merely for illustration, not for measurement), and drawing 
parallels to the abscissa line at distances from it representing 
- f-7r , - J-7T , J 73 - , -|7t , -§- 73 - , etc. A parallel at distance - £ 73 - is an 
asymptote to each of the argument-curves, and is shown in 
diagrams 2, 3, 4, on one scale of ordinates. The parallel corre- 
sponding to distance ^-7 r is shown in the fifth and sixth diagrams, 
on the smaller scale of ordinates used in their argument-curves. 
§ 108. The first diagram shows zeros at x— ± J3 , of which that 
at x = — ^3 is marked 1 . In the second diagram the argument- 
curve indicates zeros for the - and - J 7 r parallels, which are 
seen distinctly on the water-curve. The zero corresponding to 
the - J 7 r parallel was formed at the origin at the time when \gfi 
was equal to 3 , that is, when t was 1/^/2, or *707. It is a 
coincidence of two zeros for ^-positive and cc-negative. 
Diagram No. 3 shows that, shortly before its time, a maximum 
has come into existence in the argument-curve, which still indicates 
only two zeros. These are marked by crosses. 
