580 Proceedings of Royal Society of Edinburgh. [sess. 
§ 54. Given, as forcive, the isolated distribution of pressure 
defined in fig. 1 2, travelling at a given constant speed ; required 
the steady distribution of displacement of the water in the place 
of the forcive, and before it and behind it ; which becomes estab- 
lished after the motion of the forcive has been kept steady for 
a sufficiently long time. Pure synthesis of the special solution 
given in §§ 1-10 above, solves not only the problem now proposed, 
but gives the whole motion from the instant of the application 
of the moving forcive. This synthesis, though easily put into 
formula, is not easily worked out to any practical conclusion. On 
the other hand, here is my present short but complete solution of 
the problem of steady motion for which we have been preparing, 
and working out illustrations in §§ 32-53. 
Continue leftward, indefinitely, as a curve of sines, the D curve 
of each of figs. 18, 19, 20; leaving the forcive curve, F, isolated, 
as shown already in these diagrams. Or, analytically stated : — 
in (89) calculate the equal values of d(0) for equal positive and 
negative values of 0 from 0° to 40° or 50° by {(83), (86), (87)} ; 
and for all larger values of 6 take 
d(0)=( - 1) ; d(180°) sin (j + i)0 (91), 
where d(180°) is calculated by {(83), (86), (87)}. This used in 
(89), makes D(0)==O for all positive values of 0 greater than 40° 
or 50° ; and makes it the double of (91) for all negative values of 
$ beyond - 40° or - 50°. 
§§ 55, 56. Rigid Covers or Pontoons , introduced to apply the given 
forcive ( pressure on the water-surface ). 
§ 55. In any one of our diagrams showing a water-surface 
imagine a rigid cover to be fixed, fitting close to the whole water- 
surface. Now look at the forcive curve, F, on the same diagram, 
and wherever it shows no sensible pressure remove the cover. 
The motion (non-motion in some parts) of the whole water remains 
unchanged. Thus, for example, in figs. 13, 14, 15, 16, let the 
water be covered by stiff covers fitting it to 60° on each side of 
each C ; and let the surface be free from 60° to 300° in each of 
the spaces between these covers. The motion remains unchanged 
