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



ing the initial of the tide. Thus, for example, the M 2 tide is ex- 

 pressed by 



f m H m cos 



I have allowed a departure from this notation in the case of 

 the tides K 2 and K ls where I write H", *c", f" for the first, and 

 H', K', f for the second. The angles 2v" and i/ (which, like v and 

 f, are functions of N) are also involved in the arguments* (or angle 

 under the cosine in the expression for the height of the particular 

 tide) of these two tides. 



It is obviously necessary to suppose the reader to have some 

 acquaintance with the harmonic notation, or it would be necessary to 

 repeat the Report on Tides above referred to. 



3. The General Method of Treating H. and L. W. Observations. 



Noon of the day on which the observations begin is to be taken as 

 the epoch, and the mean solar time elapsed since epoch is noted by t. 

 V with the proper subscript letter denotes the increase of argu- 

 ment since epoch ; for example, F, = %(<y a)t. 



Then the height of the water h, estimated from mean sea-level, is 

 expressed by a number of terms of the form A cos F+-Bsin F, or, in 

 an alternative form, B cos (F ). 



In order to explain the principle of the method proposed, let us 

 take two typical terms involving V p and F 2 , and let the rates of 

 increase of V p be p, and of V q be q. 



Then we have 



h = Ap cos Vp + Bp sin V p +A q cos V q +B q sin V q ..... (1). 



Since at H. or L.W. h is a maximum or a minimum, we must 

 have 



= A sin VpBp cos Vp + ^-A s sin V q --B q cos V q ..... (2). 



q q 



Let us write g_ _ , 



~ 



Then multiply (1) by cos V p and (2) by sin V p , and add ; and 

 again multiply (1) by sin V p and (2) by cos V p , and subtract, and we 

 have 



h cos V f = Ap + A ? (cos V p cos V q + Jc q sin V p sin F 2 ) 

 + 2 (cos V f sin F 2 fcj sin V p cos F ff ), 



ft sin V f 



-\-Jj q \V(J8 K^jiSiu. f j n/j Dili r p \^ua t q j, 



+ A q (sin V p cos V q Jc q cos F^ sin F ? ) 

 + B 9 (sin F p sin V q + k q cos V p cos F ? ) .. 



fVint. T Vin.v snTnfifiTnfts plsfi'wlifire used arffu: 



* It is well to explain that I have sometimes elsewhere used argument to denote 

 the argument according to the equilibrium theory, that is to say, with K equal to 

 zero. In this paper I call the latter the equilibrium argument. 



