Ship Waves. 467 
the discontinuities in f’(x) at stern and bow are both positive; at an inter- 
mediate sharp corner, say, at a shoulder, the discontinuity weuld usually 
be negative. Along the curved lines of the model f(A) is negative, except for 
hollow lines where the form is concave outwards and where f’’(h) is positive. 
Thus, knowing the character of the function F, the expression (5) gives a 
general idea of the contributions of the various parts of the form. These 
possibilities are illustrated in fig. 1, which represents a half section of a model 
by a horizontal plane ; or, to pe more exact, the diagram gives the distribution 
R F (2) A 
Stern Bow 
Fic. 1. 
of doublets which is approximately equivalent to a model of this form. The 
figure also indicates the conventions for direction which are adopted throughout 
this paper. 
3. We now isolate one particular feature for examination separately. It 
should, however, be noted that the function Q defined in (3) increases without 
limit as its argument becomes greater, though the expression (5) for the model! 
as a whole remains finite everywhere. Therefore there is a certain artificiality, 
as regards that part of the disturbance, in applymg the expressions to an 
isolated element of the form; but that may be allowed for, and in any case 
the method gives the differences made by changes in any particular element. 
Consider a point on the model, given by « = a,, where the lines of the model 
are straight lines meeting at a finite angle, for example, P, Q, or R in fig. 1. 
Let C be the discontinuity in f’ (x) at that point ; that is, C is the difference 
of slope of the lines forward and aft of that point. Then, from (2) and (5), 
the contribution of this element to the surface elevation is 
a= ae (x — a) 
= (4C/rK9) {= 4£Q5 (kom) + Po (Ko7's)}; (6) 
where q, = «© — a and q’,; =«x,— «x. Further, we may use (6) for all values 
of a with the convention that P, is to be taken zero for negative values of its 
argument, and that Qo (— p) = Qo (p). Now suppose that the same change of 
slope is carried out uniformly in a given range; that is, suppose the sharp 
362 
