76 Transactions of the Royal Canadian Institute 



accord with the declination and the distance, which agree closely with 

 each other. Hence the general principle is that any missing day in the 

 gap, can be interpolated between a pair of curves, drawn on the same 

 sheet, which are for the corresponding days a month previous and a month 

 later. 



The true average lengths of the "months" with which we have to deal 

 are as follows, in mean solar days: 



Synodic month of the Moon's phases, 29.5306 days 



Anomalistic month of the Moon's distance 27.5545 " 



Month of the Moon's declination 27.3216 '■ 



In dealing with these periods, a difficulty arises at the outset because 

 of the fractional half day in the Synodic month. To obtain an interval 

 equal to the true length of that month before and after any missing 

 day, a preliminary interpolation ought first to be made between the 

 curves at 29 and 30 days before, and 29 and 30 days after; to obtain two 

 curves which represent hypothetical days between the actual days at 

 the proper interval of 29>2 days. A final interpolation between this 

 pair of hypothetical days, will then give the missing day required. 

 This method has been tried; and although tedious, it should be satis- 

 factory in dealing with tides in which all other variations are negligeable 

 except those that recur in the period of the Synodic month. But in the 

 more complex problem of Declinational tides, it was found that it was 

 not practicable to adhere strictly to the Synodic month alone; and also 

 that the double work entailed by this preliminary interpolation could be 

 obviated by a difTerent procedure. 



It was accepted as a working basis, that some compromise would 

 have to be made between the length of the Synodic and Declinational 

 months. Any variation in the Anomalistic month would thus be dealt 

 with also, as its length is so nearly the same as the Declinational. 

 The actual mean between the periods of the Synodic and Declinational 

 months is 28.43 days, or very nearly 28>^ days. The difficulty of a half 

 day reappears. Trial interpolations were accordingly made between 

 pairs of curves at equal intervals before and after any missing day, the 

 interval ranging from 27 to 30 days to cover the various months. It 

 was finally decided to adopt an interval of 29 even days before and after 

 the missing day. This gives somewhat greater weight to the Synodic 

 month, as it is only half a day short of its length ; and it is over 1}4 days 

 longer than the Declinational month. 



When pairs of curves at this interval of 29 days before and after the 

 missing day were placed together on the same sheet, there were times 

 when on the one hand, the pair might be too widely spaced in time to 



