388 Periodic Irrotational Waves at Surface of Deep Water. 



at the surface is also constant to a high degree of ap- 

 proximation. The greatest difference is ("000,001,0), which 

 may be compared with *4, the latter amount representing 

 the corresponding statical difference at the crest and trough 

 of the wave. According to this standard the pressure at the 

 surface is constant to 2^ parts in a million. 



The advantage gained by the introduction of /3 and 7 will 

 be better estimated by comparison with a similar calculation 

 where only a (still equal to j 1 ^) and g are retained. By (2) 

 in this case 



W-2gy-l = Je-*+<L(l-g)y. . . . (23) 



Table II. shows the values of ?/and of o?e~ 2u corresponding 

 to the same values of x as before. The fourth column gives 

 (23; when g is so determined as to make the values equal at 

 0°and 180°. It appears that the discrepancy in the values of 

 U 2 — 2gy is reduced 200 times by the introduction of ft and 7, 

 even when we tie ourselves to the values of /3, 7, a prescribed 

 by approximations on the lines of Stokes. 



Table II. 



cc. 



V- 



u*e- 2y. 



U2_2^_i. 







+ •091,276,5 



•008,331,4 



•010.207,7 



22^ 



•084,870,5 



008,438,8 



. . .183,4 



45 



•066,182,4 



■008,760,2 



. . . 120,7 



67i 



•036,882,6 



•009,288,9 



. . .047,1 



90 







•010.000,0 



. . .000,0 



U2J 



-•039,823,1 



•010,829.0 



. . .010,4 



135 



-•((76,318,5 



•011,6-19,0 



. . .080,2 



157| 



-•102.344,1 



•012,271.4 



. . . 167,6 



180 



-•111,832.6 



01 2,500,5 



•010,207,7 



■ i 



A cursory inspection of the numbers in column 4 of 

 Table I. suffices to show that an improvement can be 

 effected by a slight alteration in the value of /3. For 

 small corrections of this kind it is convenient to use a 

 formula which may be derived from (2). We suppose that 

 while a and f are maintained constant, small alterations S/3, 

 £7, hg are incurred. Neglecting the small variations of 

 /3, 7, g when multiplied by o? and higher powers of a, 



we get 



and 



S(IP 



&y — 8/3\ cos 2 a' — t] a cos oc — | <x cos 3* } 



+ £7(00.5 3^' -2a cos 2.1' -2a cos 4^}, • • (24) 



- 2<7//) = 2x(S/3 — Sg) cos x 4- i Sff cos 2x 

 + 2(287 — a $£) cos 3^ — Ga&ycos 4,i\ . . .(25 



