1082 Proceedings of Royal Society of Edinburgh. 
[sess. 
hollows, all shaped according to the isophasal curves shown in 
fig. 32. Looking at any one of the short arc-ridges and following 
it through the cusps, we find it becoming the middle line of a 
valley in each of the long arcs of the curve. And following a 
short arc mid-valley through the cusps, we find, in the continua- 
tion of the curve, two long ridges. Every ridge, long or short, is 
furrowed by valleys. All the curved ridges and valleys are parts 
of one continuous system of curves, illustrated by fig. 32 and 
expressed by the algebraic equation (129). 
With these explanations we may write (118)' as follows : 
d(x, y) = 
Arr 2 bk, sec 2 i f/ 
where 
XfS 
sin — l ru 
(130), 
(131). 
§ 93. An important, perhaps the most important, feature of the 
wave-system which we actually see on the two sides of the mid- 
wake of a steamer travelling through smooth water at sea, or of a 
duckling* swimming as fast as it can in a pond, is the steepness 
of the waves in two lines which we know to be inclined at 19° 28' to 
the mid-wake. The theory of this feature is expressed by the 
coefficient of the sine in (130), and is well illustrated by the 
A * S6 ^ ~ ^ or e ^ even P°i n l s °f any one of the 
curves of fig. 32, the results of which are shown in column 6 of 
the following table. They express the depression below, and 
elevation above mid-level, due to one constituent of the system of 
crossing hills and valleys described in § 92. Column 1 is -if/. 
Columns 2, 3 are xja and y/a, calculated from (127). Column 4 
is u, calculated by (126) from columns 2, 3. Column 5 is 
^ • d 2 u/dif/ 2 , calculated from (124) and columns 2, 4. Column 6 
calculation of 
"\/r 
sec 2 if/ 
calculated from columns 1, 6. 
bein 
g, as we 
* In the case of even the highest speed attained by a duckling, this angle is 
perhaps perceptibly greater than 19° 28', because of the dynamic effect of the 
capillary surface tension of water. See Baltimore Lectures , p. 593 (letter 
to Professor Tait, of date 23rd Aug. 1871) and pp. 600, 601 (letter to William 
Froude, reprinted from Nature of 26th Oct. 1871). 
