Waves at the Surface of Deep Water. 387 



The next step is the calculation of approximate values of y 

 from (11), which now takes the form 



y = - -0051 + -101,165,0 cos x 



- -005,183,3 cos 2x + -000,399,6 cos %x 



- '000,033,3 cos 4;r + -000,003,3 cos 5x. . . . (22) 



For example, when x = 0, y = '091,251,3. The values 

 of y calculated from (22) at steps of 22-^° (as in Burnside's 

 work) are shown in column 2 of Table I. 



We have next to examine how nearly the value of y 

 afforded by (22) really makes yjr vanish, and if necessary to 

 calculate corrections. To this 8 and e in (1) do not con- 

 tribute sensibly and we rind yfr= + '000,015,4 for x = 0. 

 In order to reduce yfr to zero, we must correct the value of y 

 With sufficient approximation we have in general 



o>=fy(l + T V~ y cos#)*, 



or in the present case 



, -000,015,4 



1-091 



= -•000,014,1 



so that the corrected value of y for # = is '091,237,2. 

 If we repeat the calculation, using the new value of y, we 

 find + = 0. 



Table I. 



X. 



y from (22). 



y corrected. 

 + •091,237,2 



W-2gy-\. 



Corrected 



by a/3. 







+ 091,251,3 



•010,104,9 



45 



22| 



+ •084,839,7 



+ •084,841,9 



.... 4,7 



44 



45 



+ •066,182,8 



+ 066,181,8 



. . . .4,3 



43 



67£ 



+ •036.913,1 



+ •036,915,1 



... .4,1 



44 



90 



+ •000,050,0 



+ •000,052,4 



. . . .4,2 



46 



il2i 



-•039.782,7 



-•039,780,2 



... .4,4 



47 



135 



-•076,316,2 



-•076,317,5 



... .4,3 



43 



157^ 



-•102.381,1 



-•102,395,1 



.... 4,7 



44 



180 



-•111,884,7 



-•111,907,9 



•010,105,1 



47 



In the fourth column are recorded the values of U 2 — 2gy — 1, 

 calculated from (1) with omission of S and e, and with the 

 corrected values of y. dijr/dx, d^jdy were first found sepa- 

 rately, and then U 2 as the sum of the two squares. The 

 values of /?, 7, y employed are those given in (15). (18), (20). 

 The form ot ty in (1) with these values of the constants vanishes 

 when y takes the values of the third column, and the pressure 

 * The double use of will hardly cause confusion. 



