1890.] Harmonic Analysis of Tidal Observations. 297 



The coefficients F, G, f, g, as due to the tide M 2 of speed 2(7 <r), 

 will be written with suffix ru, and as due to the tide S 2 of speed 

 2(7 /), with suffix s. The sums and means have also to be taken 

 in the two ways denoted by S and SK Hence we have altogether 

 to compute sixteen coefficients, which by an easily intelligible nota- 

 tion may be written F w <>, GL<>, . . . f,^'), g,, W) . 



In order to compute the sixteen coefficients, it is necessary to find 

 the mean cosines and sines of the four following angles, viz. : 



^F m F m , ^V m V,, and the means have to be taken in the two 

 ways denoted S and S*"". 



These means are exactly the same in form as what the means of 

 h cos and h sin (which had to be evaluated in S and S*"") would 

 be if all the heights were regarded as positive unity, irrespective 

 of whether they are H.W. or L.W. Hence the same plan of com- 

 putation serves here as elsewhere ; the plan is explained in the 

 following section. 



By comparison of equation (7) and the definitions (31) of W, X, 

 Y, Z in the last section, we have : 



x = 



Y = 

 Z = 



The four quantities A m , B m , A,, B, are known from the evalua- 

 tions of the tides M 2 and S 2 ; whence the corrections referred to in 

 7 are calculable. 



9. On the Summations. 



It will be seen from the preceding sections that sums have to 

 be found of the following functions : 



, COS Y -i COS Y -i COS j^pr 



and also of 



sin SID. sin 



It is necessary to calculate the five angles |F OT , F,, ^F OT +F,, and 

 F OT , for each tide, and the reader will easily gee, by the example in 

 the Appendix, how they may be computed with considerable rapidity, 

 by aid of an auxiliary table A. 



The computation of sines and cosines and multiplication by heights, 

 may, with sufficient accuracy, be abridged, by regarding the cosine or 



VOL. XL VIII. Y 



