448 Sir William Thomson on Stationary 



Thus, if $ be chosen at a point of the water-surface where 

 the horizontal component velocity is rigorously or approxi- 

 mately equal to U, then V is rigorously or approximately the 

 vertical component velocity at ty. Using now (14) in (9), 

 with UD/D for U , we find 



D o" D = kKUu' • • • • ^-> 



9 ~ Do 2 

 which, used in (13), gives 



TPD 



9 



(16). 



Hs V i(o 0+ %uY + -i v+U2 -^- 



Hence, when the change of level, D — D, is but small, in 

 comparison with D or D , we have 



F=J— Hi- 2+ ifV + U 2 -«Vy . • (17), 



where = denotes approximate equality. Going back to (16), 

 let ty be so chosen on the water-surface that 



j" 



(18), 



which it is clear we can do, because at a crest the first member 

 is less than the second, and at a hollow greater. When the 

 motion is infinitely nearly simple harmonic (the stream-lines 

 curves of lines), the position of J J5 thus chosen will be exactly 

 the middle between crest and hollow. When the motion is 

 anything, however great, up to Stokes's highest possible wave, 

 the chosen place of *JJ is a less or more rough approximation 

 to the mid-level point of a wave : it is always rigorously de- 

 terminate. For brevity we shall call it, that is to say a point 

 defined by (18), a nodal point. Thus, when ty is taken as a 

 nodal point, (16) becomes simplified to 



IPD 



t? 4 9 D 2 rD 



8'r i(i>o + D)U : 



iTTT-rTTTT^ ^J" " 



IV 



This expression is rigorous. In it V, which is given rigorously 

 by (14), is approximately (not rigorously) equal to the ver- 

 tical component velocity at $ : and if we suppose D given, 



