302] CASE OF INFINITE DEPTH. 547 



whence, by (4) and (6) we find, at the surface, 



A - i (gk + T'k s + 2vkma) C}... .(12), 

 ^ = -{2ivk*A+(a + 2vk*)C} ............... (13), 



where T f = T-^jp, the common factor e ikx+at being understood. 

 Substituting in (10), and eliminating the ratio A : C, we obtain 



(a + 2vk*) 2 + gk+T'k s = 4<vJ<?m ............ (14). 



If we eliminate m by means of (7), we get a biquadratic in a, 

 but only those roots are admissible which give a positive value to 

 the real part of the left-hand member of (14), and so make the 

 real part of m positive. 



If we write, for shortness, 



gk + T'k 3 = a\ vk*/<r = 6, a + 2vk 2 = xv ...... (15), 



the biquadratic in question takes the form 



(tf 2 + l) 2 =16<9 3 (#-<9) .................. (16). 



It is not difficult to shew that this has always two roots (both 

 complex) which violate the restriction just stated, and two 

 admissible roots which may be real or complex according to the 

 magnitude of the ratio 6. If X be the wave-length, and c (= cr/k) 

 the wave-velocity in the absence of friction, we have 



=v k/ c = (27rvlc) + \ .................. (17). 



Now, for water, if c m denote the minimum wave-velocity of 

 Art. 246, we find 27ri>/c m = '0048 cm., so that except for very 

 minute wave-lengths 6 is a small number. Neglecting the square 

 of 0, we have x = i, and 



a = -2vfri<r ........................ (18). 



The condition p xy = shews that 



CIA = - 2ivte/(a + 2vfr) = + 2^ 2 /o- ............ (19), 



which is, under the same circumstances, very small. Hence the 

 motion is approximately irrotational, with a velocity-potential 



(20). 

 352 



