384 Lord Rayleigh on Periodic Irrotational 



The constancy of (6) requires the annulment of the co- 

 efficients of y and of cos 2x and cos 3#?, so that 



/3=-K J 7=^, .... (7) 

 and 



^ = l_ a2 _| a 4 (g) 



The value of g in (8) differs from that expressed in equation 

 (11) of my former paper. The cause is to be found in the 

 difference of suppositions with respect to yfr. Here we have 

 taken -\|r = at the free surface, which leads to a constant 

 term in the expression for y, as seen in (5), while formerly 

 the constant term was made to disappear by a different choice 

 of ty. 



There is no essential difficulty in carrying the approxi- 

 mation to y two stages further than is attained in (5), If 

 S, e are of the 6th and 7th order, they do not appear. The 

 longest part of the work is the expression of t~ y as a function 

 of w. We get 



„ H 3a 2 , 125a* 



<j-y = l+_+— _ cos ,£{« + 2a*} 



4 64 L 



(3a 2 , 125a 4 ] 2a" Q , 125a 4 , , a . 



-f cos 2x \ — + -jo P \ ~ "q~ cos v® +" -inn cos 4#, (9) 



and thence from (!) 



12 4^ f 9a 3 . 625a 5 3«£l 



5/=-i« 2 — a 4 -f COS .r j«+-£- +-^2 ~~2~\ 



' f 1 , 4a 4 _ 1 • _ f 3a 3 , 625a 5 3a£ , ) 



— cos 2^ j i a- + -^ /S I + cos 3^ | -g- + -3347"" ~2~+ f >' f 



— tt-cos4^+ -^cos ox . (10) 



When we introduce the values of /3 and 7, already deter- 

 mined in (7) with sufficient approximation, we have 



i9« f 9a 3 , 769a 5 } 



3/=-ia 2 ~a + cos ^{*+-^- + -y92" J 



_ r a 2 ^ 11a 4 5. 9 J 3a 3 315a 5 1 



-cos2*(^+^[+cos3*{^ + — } 



— -^cos4#+-^j-cos5#, (11) 



in agreement with equations (13), (18) of my former paper 

 when allowance is made for the different suppositions with 

 respect to ty, as may be effected by expressing both results 

 in terms of a, the coefficient of cos x, instead of a. 



