227] DISCUSSION OF THE INTEGRALS. 399 



The functions Ci u and Si u have been tabulated by Glaisher*. It appears 

 that as u increases from zero they tend very rapidly to their asymptotic values 

 and ^?r, respectively. For small values of u we have 



where y is Euler s constant -5772.... 



The expressions (19), (20) and (21), (22) alike make the eleva 

 tion infinite at the origin, but this difficulty disappears when the 

 pressure, which we have supposed concentrated on a mathematical 

 line of the surface, is diffused over a band of finite breadth. In 

 fact, to calculate the effect of a distributed pressure, it is only 

 necessary to write x x for x, in (21) and (22), to replace Q by 

 Ap/p . x , where Ap/p is any given function of # , and to in 

 tegrate with respect to of between the proper limits. It follows 

 from known principles of the Integral Calculus that, if Ap be 

 finite, the resulting integrals are finite for all values of x. 



If we write x (u) = Ciu sin u + (-|7r Si w) cos u ( x ii)j 



it is easily found from (19) and (20) that, when ^ is infinitesimal, we have, for 

 positive values of #, 



no T 00 



/ ydk Sroos*r+x(i?) (xiii). 



V J x 



and for negative values of x 



-n- x - K x} (xiv). 



In particular, the integral depression of the free surface is given by 



and is therefore independent of the velocity of the stream. 



By means of a rough table of the function x (u\ it is easy to construct the 

 wave-profile corresponding to a uniform pressure applied over a band of any 

 given breadth. It may be noticed that if the breadth of the band be an 

 exact multiple of the wave-length (2r/ic), we have zero elevation of the surface 

 at a distance, on the down-stream as well as on the up-stream side of the 

 seat of disturbance. 



* &quot; Tables of the Numerical Values of the Sine-Integral, Cosine-Integral, and 

 Exponential-Integral,&quot; Phil Trans., 1870. The expression of the last integral in 

 (22) in terms of the sine- and cosine-integrals, was obtained, in a different manner 



from the above, by Schlomilch, &quot; Sur l inte&quot;grale d6finie / dd e~ X& &quot; Crelle 



6 2 + a * 



t. xxxiii. (1846) ; see also De Morgan, Differential and Integral Calculus, London 

 1842, p. 654. 



