$28 REPORT— 1844. 



as the velocity per second which a heavy body will acquire in falling freely 

 by gravity through a space equal to half tlie depth (reckoned from the crest 

 of the wave), is that given in column F ; with which the numbers in column 

 E resulting from observation are compared, their excess or defect being set 

 down with the signs + or — in column G. 



We are thus enabled to compare the numbers given by observation E 

 with the numbers given by formula F, and the result G shows that the 

 coincidence is as close as the means of observation would admit. It was not 

 possible with the chronometer then applied (although observations to fifths 

 of a second have since been obtained) to depend upon accuracy to more mi- 

 nute intervals than half-seconds, and the differences in column G are precisely 

 what we should have expected, being nearly alternately + and — , and being 

 of nearly the same magnitude at both ends, and along the whole line of ob- 

 servation. The sum of the errors affected by the positive sign is -|-r36, 

 the sum of those affected with the negative sign — O'S't, so that the whole 

 of 29 observations give only an excess of +•52, or a mean excess of O'OIS, 

 showing a mean excess of velocity of the observation over the velocity assigned 

 by the formula, of 0"018 of a foot per second, being less than ooTT^^ o*' ^''^ 

 whole. Hence we are warranted in assuming, that as far as the history of 

 this wave is concerned, the velocity is accurately represented to within ^uu*''^ 

 part by the formula '^ g(Ji. + k)=v. 



Experiments on the Velocity. — In order to determine the velocity of the 

 wave of the first order with accuracy, a series of experiments have been made 

 upon rectangular channels, extending from 1 inch in depth and a foot wide, 

 to 12 feet wide and 6 feet deep. These experiments, forming a series of thirty 

 different depths, are given in Table III. Column A contains the depth of the 

 water, reckoned from the crest of the wave. Column B is the height of the 

 crest of the wave above the level of the water in repose. Column C is the 

 velocity of the wave as observed, and in column D is given the vel ocity du e 

 to half the depth in column A calculated by the formula t?= \^ g{h-\-k). 

 Columns D and C are compared, and their difi'erence given in E, from which 

 it results that the formula represents the experiments to within a mean error 

 of 0-007. The results of this table leave no room to doubt that, as far as 

 observation can settle this point, the velocity is conclusively settled, and de- 

 termined to be that due hy gravity through Jialfthe depth ofthejluid, reckoned 

 from the ridge of the wave. 



