576 Proceedings of Royal Society of Edinburgh. [sess. 
(77), v = Jga/^TT, and X = «/4*5. Kemark that the scale of 
ordinates is, in fig. 15, only 1 /2 *5 of the scale in fig. 16 ; and see how 
enormously great is the water-disturbance now in comparison with 
what w T e had with the same forcive, but three times greater speed 
and nine times greater wave-length ( v = Jga/Tr, \ = 2 a). Within 
the space-period of fig. 15 we see four complete waves, very approxi- 
mately sinusoidal, between M, M, two maximums of depression 
which are almost exactly (but very slightly less than) quarter 
wave-lengths between C and C. Imagine the curve to be exactly 
sinusoidal throughout, and continued sinusoidally to cut the zero 
line at C C. 
We should thus have in CC a train of 4J sinusoidal waves; 
and if the same is continued throughout the infinite procession 
.... C C C .... we have a discontinuous periodic curve 
made up of continuous portions each 4J periods of sinusoidal 
curve beginning and ending with zero. The change at each point 
of discontinuity C is merely a half-period change of phase. A 
slight alteration of this discontinuous curve within 60° on each 
side of each C, converts it into the continuous wavy curve of fig. 15, 
which represents the water-surface due to motion at speed JgajQ tt 
of the pressural forcive represented by the other continuous curve 
of fig. 15. 
§ 49. Every word of § 48 is applicable to figs. 14 and 13 except 
references to speed of the forcive, which is ^ga/ld-n- for fig. 14 
and Jga/ 4l7r for fig. 13; and other statements requiring modifica- 
tion as follows : — 
For 4J “periods” or “waves,” in respect to fig. 15; substitute 
9J in respect to fig. 14, and 20 \ in respect to fig. 13. 
For “depression” in defining M M in respect to figs. 15, 14; 
substitute elevation in the case of fig. 13. 
§ 50. How do we know that, as said in § 48, the formula 
{(83), (86), (87)} gives for a wide range of about 120° on each 
side of 0= 180°, 
d(0)=(-l)'d(18O'). sin(i + |)0 . . . (88), 
w T hich is merely §§ 48, 49 in symbols 1 it being understood that j 
is any integer not < 4 ; and that e is *9, or any numeric between 
•9 and 1 ? I wish I could give a short answer to this question 
