48 REPORT — 1886. 



Next, if we put 



a = C,H" cos (<C"- O, /3 = C2H" cos (K"-iC„), 



A=acos2A , B=/3cos2A (14) 



then obviously «, /3 are absolute constants for the port, and A and B are 

 nearly constant, for their small variability only depends on the longitude 

 of the moon's node entering through A. 



Thus we have, from (9), (12), (13), (14), 



M=fH,„+(p'-l)fH,„ + («cos2o'-A), 



^ =!,•,„ + (/3 cos 2o' — B), expressed in degrees, 



M=l-086p,cos2a,H3, 



F.='^s (15) 



where p', o' are the values of p and 2 at a time earlier than that corre- 

 sponding to ;// by ' the age ' 52''-2 tan ((," — )>•,„). 



In the article fH,„ is called R.^ ; (p' — l)fHin, the parallactic correction, 

 is called c^B,^ ; (a cos 23' — A), tlae declinational correction, is called BjHm- 

 Similarly, /3 cos 25' — B, the declinational correction to Knj, is called 22'''m- 

 Also, My is called S. 



Thus, with this notation the whole semidiurnal tide is 



7i2=(Il^ + aiR^+a2EJcos(2i^-.-„-cV,J + Scos(2^,-.-3) . (16) 



The mean rate of increase of \p is y — o-, or 14-°49 per hour; hence 

 the interval from moon's transit to lunar high water is approximately 

 ^^(i^m + ^-i'^m) bours, when )>■„ is expressed in degrees. If i be the mean 

 i'nterval, and ()2i its declinational correction, 



i + ^2^=2V'^*m+aV^2'^'m (17) 



Now let A be twice the apparent time of moon's transit reduced to 

 ancle at 15° per hour, or the apparent time reduced at 30° per hour, 



Then the excess of the moon's over the sun's R.A. at lunar high water 

 is iA plus the increase of the difference of R.A.'s in the interval i. This 



increase is approximately ^^^''^:,n, and at lunar high water the sun's hour- 



angle is given by 



2v/.,=2>^ + ^+^--\„ (18) 



•V — a 



Since the difference of time between lunar high water and actual high 

 water never exceeds about an hour and a half, if we neglect the separation 

 of the moon from the sun in that time, this relationship also holds at 

 actual luni-solar high water. 



Now, let 



Hcos (u — d))=M+Scos [4 + -~')c,„-r-(.-g + (.-^ + a2':m] 



^' ' y — (T 



=M+Scos(4 — (Cs + t^K-m + OaOj 

 Hsin(f«-^)= Ssin (4-(.-3 + |f(.-n, + Vm) .... (19) 



and we have for the whole luni-solar semidiurnal tide 



7i2=Hcos(2;//-^) (20) 



