J-> 



548 Wave-Patterns due to a Travelling Disturbance. 



where r= \/(^ 2 + ?/ 2 ), we have, by differentiation with 

 respect to b, 



it = ^ - r5 n 



cos ty + i sin ^ + ibf (r 2 + b 2 )V 



Inserting the various constant factors that have been 

 omitted, we find that the part of the value of f which ha& 

 been omitted has the value 



2^p(r 2 + b 2 )l ( 52 > 



This is negligible when the distance r from the origin m 

 large compared with b. 



8. When the velocity c of the travelling agent is sufficiently 

 small the influence of capillarity predominates over that of 

 gravity. Taking account of it alone we have 



* = ^rcos 2 0, (53) 



where T is the surface-tension. The forms of the isophasal 

 lines are accordingly determined by 



P=^^0, ..... (54). 



giving a quasi-parabolic shape. Since 



U=|ccos0, (55) 



the formula (27) is now to be replaced by that derived from 

 (21) in a similar manner. The approximation makes £* 

 vanish unless #cos0 + ^sin# in negative, so that the head 

 (0 = 0) of the wave-system lies to the left of 0, i. e. it 

 precedes the disturbing agent. 



Again, by (35) we have, along an isophasal, 



/"((9)=-6n7r(l + 2tan 2 (9). . . . (56), 



The final result takes the comparatively simple form 



£= <A 4/ g C57V 



& VnirT* v / (3 + 6tan 2 (9) '* { y 



