428 Proceedings of Royal Society of Edinburgh. . [sess. 
§ 143. The £ 4 ,( 0 , 1 , t) curve of fig. 38 may be compared with 
the curve of the same designation in fig. 36. They differ because 
of a quarter period difference in the phase of commencement of 
the disturbing pressure, which commences suddenly at its 
maximum for all the curves of fig. 36, and commences at zero 
for all the curves of fig. 38. If the , S$. curves for initiating 
pressure commencing at zero were drawn, they would differ from 
the first and third curves of fig. 36 in being at the commencement 
tangential to the line of abscissas, instead of being inclined to it 
in the positive direction, as shown in fig. 36. The £ curves are 
all initially tangential to the line of abscissas, but the tangency 
is only of the first order in fig. 36, while it is of the second order 
in fig. 38. 
§ 144. The third and fourth curves * of fig. 38 show the whole 
history for the points, x = 0 , and x = A. , of the surface displacement 
expressed by the formulas 
^(x , 1 , t) = - f dq sin (o(t - 2 )-ife , 1 , q) , (191), 
9J 0 ay 
which expresses the surface displacement due to surface-pressure 
expressed by 
H(x , 1 , t) = - sin wt i]/(x , 1 , 0) . . . (192). 
The fifth curve of fig. 38 shows the history, after period 3, to 
almost half a period after period 9, of the disturbance at the 
place x = 32. The disturbance has not yet become sinusoidal, but 
would certainly become almost exactly sinusoidal after a few more 
periods. 
§ 145. In fig. 39, two sets of five curves show, for case <f> and 
case i(/, the periodically varying water-surface on each side of the 
middle, at any long enough time after the beginning of the 
motion, to give a regular regime of sinusoidal vibration as far as 
two or three wave-lengths on each side of the middle. The third 
curve in each case is a curve of sines. The first curve represents 
the surface at the beginning of. a period from it to (i + l)r. The 
fifth curve, being the first curve inverted, re {>resents the water- 
surface at the middle of the period. The other two curves may 
* The scale of ordinates of the third, fourth, and fifth cui ves ‘of fig. 38 is 
double that of the first and second, indicated on the figure. 
