HARMONIC ANALYSIS AND PREDICTION OF TIDES. 95 



o 



Let - = the acceleration in the principal component A due to the 

 disturbing component B. Then for a maximum of (366), 



a 

 This value of t must satisfy equation (368), therefore we have 



Aa sin i2nT-d)+Bl> sin F- (2n tt - - a) -f /3 

 = -Aa sin d-^BhsmP-^{2nTr-e-ay+l^-a-d~\^0 (371) 



At the time of this maximum, when 



2nT — a — d 



a 

 the phase of component A will equal 



(2n. 'ir-a-d)+a 

 and the phase of component B will equal 



Let = phase of component 5 — phase of component A at this time. 

 Then 



= ^:i^' (27i7r-a-0)+/?-a; (372) 



Substituting the above in (371) 



— Aa sin d + Bh sin {(t> — d) 

 = — Aa sin d + Bh sin ^ cos 6 — Bl cos ^ sin 

 = - {Aa + Bh cos (/>) sin d + Bh sin cos = (373) 



Then 



Bh sin cj) rnnA\ 



tan^ = -T — , -Q-L , (374) 



Aa-r-Bb COS*/) 



For the resultant amplitude at the time of this maximum substitute 

 the values of t from (370), in (366), and we have 



y = A cos {2mr-d)+B cos\-{2mr-e-a)+p 



= A cosd + B cos[^^ {2nT-d-a)+l3-a-dl 



= Acosd + B cos {(p - e) (375) 



= A cos d + B cos <t> cos 5 + 5 sin sin 6 

 = ( J. + 5 cos 0) cos d + B sin 4) sin 8 



= V^' + 5^ + 2^5 cost/, cos (5- tan-i 5 sm </> \ 



\ -d + ij cos </)/ 



