278 Prof. G. H. Darwin. [June 19, 



XL " On the Harmonic Analysis of Tidal Observations of High 

 and Low Water." By G. H. DARWIN, F.R.S., Plumian Pro- 

 fessor and Fellow of Trinity College, Cambridge. Received 

 June 17, 1890. 



1. Introduction. 



Extensive use of the tide-gauge has only been made in recent years, 

 and by far the largest number of tidal records consist only of obser- 

 vations of high and low water (H. and L.W.). Such observations have 

 usually been reduced by determining the law governing the relationship 

 between the times and heights of H. and L.W. and the positions of the 

 moon and sun. This method is satisfactory so long as the diurnal 

 inequalities are small, but it becomes both complex and unsatisfac- 

 tory when the diurnal inequality is large. In such cases the harmonic 

 notation for the tide is advantageous, and as, except in the North 

 Atlantic Ocean, the diurnal inequality is generally considerable, a 

 proper method of evaluating the harmonic constants from H. and 

 L.W. observations is desirable. 



The essential difference between the method here proposed and that 

 followed by Laplace and his successors is that they introduced astro- 

 nomical considerations from the first and applied them to each H. and 

 L.W., whereas the positions of the sun and moon will only be required 

 here at a single instant of time. In their method, the time of 

 moon's transit, and hence the interval, was found for each tide ; the 

 age of the moon, and the moon's and sun's parallaxes and declinations 

 were also required. An extensive table from the astronomical 

 ephemeris was thus necessary, and there still remained the classifica- 

 tion of heights and intervals according to the age of moon, and two 

 parallaxes, and two declinations. The classification could hardly be 

 less laborious, and was probably less mechanical, than the sorting 

 processes employed below. There is probably, therefore, a consider- 

 able saving of labour in the present method, and, besides, I conceive 

 that the results are more satisfactory when expressed in the harmonic 

 notation. 



My object has been to make the whole process a purely mechanical 

 one, and, although nothing can render the reduction of tidal observa- 

 tions a light piece of work, I believe that it is here presented in a 

 form which is nearly as short as possible. 



The analytical difficulties to be encountered in such a task are 

 small, but the arrangement of a heavy mass of arithmetic, so as to 

 involve a minimum of labour and therefore of expense, is by no means 

 easy. How far I have succeeded must be left to the decision of those 

 who will, I hope, use the methods here devised. 



