850 Prof. G. H. Darwin. On an Apparatus for [Dec. 15, 



Before entering on the details of ray plan it is proper to mention 

 that Dr. Borgen has devised and used a method for attaining the 

 same end. He has prepared sheets of tracing paper with diagonal 

 lines on them, so arranged that when any sheet is laid on the copy of 

 the observations written in daily rows and hourly columns, the 

 numbers to be summed are found written between a pair of lines. 

 This plan is excellent, but I fear that the difficulty of adding correctly 

 in diagonal lines is considerable, and the comparative faintness of 

 figures seen through tracing paper may be fatiguing to the eyes. Dr. 

 Borgen's plan is simple and inexpensive, and had I not thought that 

 the plan now proposed has considerable advantages I should not have 

 brought it forward. 



In the investigations which follow the notation of 'the Report of 

 1883 to the British Association on harmonic analysis is used without 

 further explanation. 



2. Evaluation of A , Sa, Ssa, S lf S 2 , S 4 , S 6 , T, R, K 2 , K,, P. 



The 24 mean solar hourly heights of water are entered in a 

 schedule of 24 columns, with one row for each day, extending to n 

 days ; the 24 columns are summed, and the sums divided by n ; the 

 24 means are harmonically analysed ; it is required to find from the 

 results the values of the harmonic constituents. 



The speed of any one of the tides differs from a multiple of 15 per 

 hour by a small angle ; thus, any one of the tides is expressible in the 

 form Hcos [(15gr )*], where q is 0, 1,2, 3, &c., and ft is small. 



When t lies between O h and 24 h this formula expresses the oscilla- 

 tion of level due to this tide on the day of the series of days. 



If multiples of 24 h from 1 to n 1 be added to t, the expression gives 

 the height at the same hour, , of mean solar time on each of the suc- 

 cession of days. 



Then if |j denotes -the mean height of water, as due to this tide 

 alone, at the hour t, we have 



-* 24 *- 



CD. 



When t is put successively equal to O h , l h , ____ , 23 h we get the 24 

 values of $ which are to be submitted to harmonic analysis. 



The mean value of $, say A (not to be confused with A as written 

 at the head of this section, where is denoted the mean sea level above 

 datum) is found by taking the mean of the 24 values of Jj. 



By the formula for the summation of a series of cosines it is easy to 

 prove that 





