1892.] facilitating the Reduction of Tidal Observations. 371 



The algebraical steps involved in the evaluation of these four 

 integrals and subsequent simplification are omitted. 

 Hence the result is 



cos n (0 f (a o)). 



By reference to the figure it is clear that 

 i + 5-fft =15, ft = 24e, a =7^ llje, 1} = Y^ 12^-e, 



Write, then, 



j. 12 ne /-n 



sin!2ne sin- 1 ^-^ 



and we obtain as the mean value of cos nO, when found in this way, 



JT cosw (6 f e). 



It is obvious that if we had begun with sin nO, the argument in the 

 result and the factor J^ would have been the same. Accordingly, a func- 



R' 



tion R'cos (nO ') would yield the result cos [n(6 e) f ]. 



f 



If 24 equidistant results of this sort are submitted to harmonic 



analysis to find A M , B w , we shall get 



R' 



AM = cos ( +f ne) = R cos ", suppose, 



B M = - sin (^'+4 ne) = R sin , suppose. 



R' 

 Accordingly R = -JT ^ = S" + J we. 



But it is required to find R x , ^', so that 



Thus when th.e 24 observed hourly tide heights on any m.s. day are 

 regrouped so that the observed height at 12 h m.s. time is reputed to 

 appertain to an exact special hour, and each of the previous and sub- 

 sequent hourly values of that m.s. day are reputed to belong to previous 

 and subsequent exact special hours ; and when a long series of m.s. 

 days are treated similarly, and when the mean heights of water at 

 each of the 24 special hours are harmonically analysed, we shall obtain 

 the required result by augmenting R by a factor <Jf w , and by subtract- 

 ing f ne from . 



