261 



increased or decreased by the constant values of any of its overtides 

 at these instants (par. 78). As the overtides are relatively small, the 

 mean value of TH may be taken, for purposes of computing a cor- 

 rection, as M2. A^/ is always positive, whether high water occurs 

 before or after the high water of the principal component. In the 

 long run, for every value of A?/ occurring when high water is in the 

 lead, an equal value will occur when high water lags behind. Neg- 

 lecting the effect of overtides, the height of mean high water above 

 mean sea level is therefore the amplitude, M2, of the principal com- 

 ponent plus one-half of the numerical mean value of Aij. 



4. Representing, for generality, the ordinate of the dominant com- 

 ponent as A cos (a^+a), and the ordinates of the other components 

 as Bi cos (61^+ ft), Bi cos (62H-1S2), etc., the equation of the tide takes 

 the form: 



2/=^cos (a^ + a)+^i cos (6i^+/3i)+52COs (62^ + 132)+ • • • (lA) 



Since ^y is the change in y due to a relatively small increase, A^, in t, 

 its value is approximated by differentiating the right-hand member 

 of equation (1), and is: 



^y= 



-[Aa sin {aU+a) +BJ), sin (5i^o+ft) +^262 sin (60/0+182+) ' ' ' ]Af (2A) 



in wdiich U is a time at which the ordinates of the dominant com- 

 ponent is a maximum. Such times occur when ai+a:=0, 2-n-, 4:ir, Qtt, 

 etc. The value of t^ is given by the equation: 



atQ-\-a—2mr 

 whence: 



U = 27ix/a — aja (3 A) 



where n is any integer. 



Substituting this value in equation (2A) : 



Ay= — [^a sin 2mr-\-Bihi sin (2mrbi/a^abi/a-\- (3i) 



.+^262 sin (2??.7r62/tt— a62/a+i82)+ * ' ' W. (4A) 



Since the generating radius CP of the dominant component moves 

 through the angle v with the speed a in the time At: 



At=av 

 Placing for convenience 



2mrbila—abi/a-{- ^i=Xi, 2mrb2la— ab^la-^- ^2=^2, etc. (oA) 



Then, since sin 2n7r=0, equation (4A) reduces to: 



Ai/= — [5i&i sin .ri+5262 sin J2 " ' ' ]«2'- (6A) 



