E. W. Brown — Apparatus for Tidal Analysis. 389 



The other rearrangements follow a similar process ; a separate 

 set of instructions for the ordering of the bands is given to 

 the operator for each arrangement. After the process is com- 

 pleted the cards are slipped out of the covers carrying the 

 bands, and the latter can be stored away in an envelope on 

 which it is only necessary to write the year and the port. The 

 observations have thus only been written once and are always 

 available for future reference. 



If the observations be typed on to the sheets, the dimensions 

 of the apparatus may be conveniently reduced in the ratio 3 :.2. 



It remains to explain the instructions to be given to the 

 operator. The single and double pairs of red lines and the 

 number of the day are the guides; one pair and one day num- 

 ber will be found on each face of the card when the bands are 

 placed over it. Define 'no step' for any day as a case when 

 the observation at h on its band is immediately under that for 

 h on the band next above, and 'one step left' when that at l h 

 is immediately under that at h on the band next above. Simi- 

 larly for 'one step right'. The words 'left 1 and 'right' need not 

 be repeated since there are never both left and right steps with 

 any one tide. The instructions to the operator consist only in 

 giving to him the step for each day, and some other fact which 

 will enable him to test his work. The method of obtaining 

 the instructions and the test will be explained by giving in 

 detail those for what Darwin calls 'mean lunar time'. 



The speed of this time in degrees per mean solar hour is 

 14°-4920521. It therefore moves 347°*80925 in 24 hours. 

 This, on division by 15, shows that 23*187283 mean lunar 

 hours are equivalent to 24 mean solar hours. As we can only 

 use observations at exact solar hours, the position of the band 

 on day n is obtained by finding the nearest integer to 



(n + 1)23-187283, 



the approximate coincidence being made in the middle (12 h ) 

 of each day of observation. Thus, for days 1, 2, 3, 4 the red 

 lines must be one step left of those on the previous days, 

 respectively, for day 5 no step, for days 6, 7, 8, 9, 10 one step 

 left, for day 11 no step, and so on. The whole series is ob- 

 tained by converting -187283 into a continued fraction. The 

 successive convergents are 



l_ , 2 , 3 , 53 , 162 , 



5 11 16 283 865 



The third convergent shows that the successive 6th, 5th, 5th 

 days are to be no-step days, while the fourth convergent shows 

 that for 283 days there must be 18 cases in which the no-step day 

 is the 6th and 35 cases in which it is the 5th (18 4- 35 = 53, 



