1917] AT THE SUEFACE OF DEEP WATER 483 first. If the series are to be depended upon, we must clearly take a smaller value. I have chosen a = -fa, and this makes by (15), (18), (20) £ = - -000,052,42, 7 = '000,000,976, g = "989,736,92 ....... (21)* The next step is the calculation of approximate values of y from (11), which now takes the form y = ~ '0051 + 101,165,0 cos os - '005,183,3 cos 2a? + '000,399,6 cos 3a? - -000,033,3 cos 4« + '000,003,3 cos ox ................ (22) For example, when an = 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 i/r vanish, and if necessary to calculate corrections. To this 8 and e in (1) do not contribute sensibly and we find ty = + '000,015,4 for a? = 0. In order to reduce i|r to zero, we must correct the value of y. With sufficient approximation we have in general ty % (1 + TO- e~y cos *')f; or in the present case so that the corrected value of y for x = 0 is '091,237,2. If we repeat the calculation, using the new value of y, we find -^ = 0. TABLE I.J .T y from (22) y corrected U*-2gy-l Corrected by dp 0 + '091,251,3 ' + 091,237,2 010,104,9 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 46 112* - -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 [* With the corrections specified in the footnote on p. 482 \ve have 7 = -000,000,905, g = -989,737,42. W. P. S.] t The double use of S will hardly cause confusion. [J With the corrections specified in the footnotes on pp. 481, 482, and calculating direct from (2), with the inclusion of the term 65e~4w cos 4#, I find that the first 5 figures in the value of I72 - 2<jy -1 are as in the table, whilst the last 2 figures, proceeding in order from £p=0 to a; = 180, become 45, 45, 44, 43, 42, 42, 45, 51, 53; after making 6 modifications in "y corrected" (third column), the first 6 figures of which remain as printed, whilst the last becomes, taken in the same order, 1, 9, 9, 1, 4, 3, 6, 3, 8, these modified values of y in every case reducing \j/ to zero to 7 places of decimals. W. F. S.] 312s X CL cos'x; ........................... (8)