386 Lord Rayleigh on Periodic Irrotational 



If we use only a, none of the cosines can be made to 

 disappear, and the value of g is 



<7=l-a 2 ~2a 4 -7« 6 (14) 



When we include also /3, we can annul the term in cos 2x 

 by making 



--&+¥) <»> 



and with this value of /3 



2 5a* 619a 6 



^ = l_ a 2_ _______ ^ 



But unless a is very small, regard to the term in cos 3a? 

 suggests a higher value of j3 as the more favourable on the 

 whole. 



With the further aid of y we can annul the terms both in 

 cos 2x and in cos 'doc. The value of /3 is as before. That of y 

 is given by 



a 5 /-, 139a 2 \ 



^=i2l 1+ ^r> < 17 > 



and with this is associated 



1 2 5a* 157« 6 

 0__l_o*-_~.__ (18) 



The inclusion of 8 and e does not alter the value of g in 

 this order of approximation, but it allows us to annul the 

 terms in cos 4% and cos 5 oc. The appropriate values are 



»— £ ^m < 19 > 



and the accompanying value of 7 is given by 



^i2( 1+ '"ir-> (20) 



while j3 remains as in (15). 



We now proceed to consider how far these approximations 

 are successful, for which purpose we must choose a value 

 for a. Prof. Burnside took a=£. With this value the 

 second term of f3 in (15) is nearly one-third of the first 

 (Stokes') term, and the second term of 7 in (20) is actually 

 larger than the first. If the series are to be depended upon, 

 we must clearly take a smaller value. I have chosen «=^, 

 and this makes by (15), (18), (20) 



£=-•000,052,42, 7 --000,000,976, # = '989,736,92. (21) 



