Deep Water due to Motion of Submerged Bodies. 63 



provided (d s #/ds 2 ) is large compared with the greatest range 

 of (5 — s ) to be considered. This implies a large number of 

 •oscillations from positive to negative in the sine term in (1) 

 within the effective range of s — s so that the limits of a 

 may be taken to be + x> with only slight error, while the 

 amplitude term may be considered constant and may be put 

 -equal to its value at s = s . Under these conditions the 

 -integral (1) becomes 



sin (0 O 4 <7 2 ) 



d<T \ 



~d 2 d 



x/tt sin (e Q + ?\ I . . (4) 



J s=s 



The value or values of s Q to be taken are determined by the 

 roots of (d0/~ds)=O i and, corresponding to each root, a term 

 similar to (4) must appear in the expression for the value of 

 the integral. When two roots of this equation coincide, then 

 (dV/'ds) and (d 2 0/ds 2 ) vanish simultaneously and we must 

 take, instead of (3), 



0+ ~ 3! dV ... (5) 



This gives us finally for (1) 



-K/i" ' 



£•¥** •••■'«» 



-ds* ) S = S 



Equation (3) cannot be employed when (d^d/'ds 2 ) is small 

 or zero. The two conditions (^0/ds) = O = (d 2 6/ds% or 

 (dp/'ds) = = (d 2 pl'ds 2 ) in the optical case, are the conditions 

 that two rays coincide at the point P. The effective 

 elements are then in the neighbourhood of a point of 

 inflexion on the wave-surface S, and the point P at which 

 the rays coincide must lie on the wave-caustic, which 

 corresponds to a line of cusps in the wave-system. 



