50 EEPORT — 1886. 



then, in accordance with Schedales B of the Report of 1883, the expres- 

 sions for the three tides are 



O =f„H„ cos (V„-0, 

 K,=f'H'cos(V'-0, 

 P =-HpCos[V'-.'-(2/i-r') + (K'-s)] (22) 



We have already seen in (6) and (7) that 



K,+P=R' cos (V'-k' + 0), 

 where 



tan^= sin (2 0-./) ^.^3f-cos( 2 -Q g, 

 ^ 3f'-cos(2 -.'')' 3cos0 



and © denotes the sun's mean longitude at the middle of the short period 

 under consideration. 



Then, if we write foHo=Ilo5 ^^^ diurnal tides, reduced to two, are 



0=Ro cos (V„— k-J, 



Ki+P=R'cos (V'-/.'-^0 • ..... (23) 



<p and R', having a semi-annual inequality, may be taken as constant for 

 about a month, but must be recomputed for each month. 



Now, suppose that we compute Vo and V at the epoch, that is, at the 

 initial noon of the period during which we wish to predict the tides, and 

 with these values put 



<|fo^/v-o— Vo at epoch, 



4^'=//— ^— V at epoch, 



then the speed of Vo is y—2tT, or 13°-94 per hour, or 360''— 25°-37 per 

 day ; and the speed of V is y, or 15°04 per hour, or 360°*986 per day. 

 Hence, if t be the mean solar time in hours on the (/i + l)th day since the 

 epoch, 



Vo-'.-o=360°w + 13°-94t-<ro-25°-37ji, 



V' + ^-..-'=360°n + 15°-04t-^' + 0°-986H. 



Therefore the diurnal tide at the time t hours on the (/( + l)th day is 

 given approximately by 



0=Ro cos [14°t— i:o-25J° X ?t], 

 Ki+P=R'cos[15"t-^' + l°xw] (24) 



If we substitute for t the time of high or low water as computed simply 

 from the semidiurnal tide, it is clear that the sum of these two ex- 

 pressions will give us the diurnal correction for height of tide at high or 

 low water. 



If we consider the maximum of a function, 



A cos 2n(t—a) + B cos n'(t— /3), 



where n is nearly equal to it', we see that the time of maximum is given 

 approximately by t=a, with a correction 3t determined from 



—2An sin (2nSi)—n'B sin n'(i—S)=0, 



or 



., 180 7i'B . ., ,,s 

 ot=— — — - sm w(t — p). 

 4<Trn nA 



