526 Sir William Thomson on Stationary 



have, by the well-known definite integral, first, I believe, 

 evaluated by Gauchy, 



^l^-^^TT •■ ■ ■ W> 



where <r 1 arid N x are given by (32) and (31). 



It is to be remarked that, inasmuch as (38) has the same 

 value for equal positive and negative values of x, the evalua- 

 tions expressed in (39) and (41) are essentially discontinuous 

 at#=0; and when x is negative, — x must be substituted 

 for x in the second member of the formulas. I hope in 

 Part IV. to give numerical illustrations ; but with or without 

 numerical illustrations, the analytical formula (39), with (41) 

 for its first term and the sign of x changed throughout when 

 x is negative, is particularly interesting as a discontinuous 

 expression for a curve passing continuously from one to the 

 other of the two curves 



ff= , 2 L,, . o-iNi sin -jj- for large positive values of xy 

 and iA - D ^ | (42). 



y= — i t)/a • fi^i sin TJ~for large negative values of x] 



For the case of b > D every term of (39) is real, and (re- 

 membering that the sign of x is changed when x is negative) 

 we see that it makes h equal for equal positive and negative 

 values of x, and diminish asymptotically to zero as x becomes 

 greater and greater in either direction. It expresses unam- 

 biguously the solution (clearly unique when b > D) of the 

 problem of steady motion of water in a uniform rectangular 

 canal interrupted only by a single ridge of magnitude A 

 across the bottom. This is the case of velocity of flow greater 

 than that acquired by a body in falling through a height equal 

 to half the depth. 



It is otherwise in respect to uniqueness of the solution when 

 the velocity of flow is less than that acquired by a body in 

 falling through a height equal to half the depth (b < D). For 

 this case the formulas (39) and (41) express a particular 

 solution of the problem of steady motion through a rectan- 

 gular canal, when regularity of the canal is only interrupted 

 by the single ridge of magnitude A. But we clearly have 

 an infinite number of solutions of this problem ; because in 

 still water in a canal of depth D we can have free waves of 

 any velocity from zero to V^/D, which is the velocity of an 

 infinitely long wave in water of depth D. In our flowing 

 water then superimpose upon the solution (39) (41), any 



