202 



An expression for v remains to be found, 



5. The maximum values of y, equation (lA), occur when dy/dt~0, 

 or when 



— Aa sin {ati + a) -BA sin {bit+^i)-B2b2 sin (bihi- ^2)- ' * • =0. (7A) 



At these maxima, the radius vector, CR\ is so close to the Y axis 

 that the angle R'CP' may be taken as equal to PCP'=v. Since the 

 generating radius of the dominant component is at CP' at the maxi- 

 mum values of y, 



ati-{-a=2n-K-\rV. 

 whence 



ti=2mr/a~ala-\-v/a. (8 A) 



Substituting this value in equation (7A) : 



Aa sin {2nT-\-v)-\-Bihi sin {2mrbi/a— abi/a-\-vbi/a-\- ^i) 

 +B2b2sin {2mrb2la—ab2la+vb2la+^2)+ ' ' • =0. (9 A) 



The first term in equation (9A) reduces to Aa sin v. Simplifying 

 the remaining terms by substituting Xi, X2, etc., for the equivalent 

 expressions given in equation (5A), the equation reduces to: 



Aa sin v-\-Bibi sin (:Xi + biv/a)-\-B2b2 sin {x2-\-b2v/a)-'r " ' " = 



Expanding the sine functions: 



Aa sin v-\-Bibi sin Xi cos vbxla-\-Bib\ cos Xi sin vbxja 



-\-B2b2 sin X2 cos ?'62M+52&2 cos X2 sin vb2la-\- ' ' " = (lOA) 



The fractions bi/a, bila, etc., are the ratios of the speeds of the various 

 components to that of the dominant component. For semidiurnal 

 components these ratios are close to unity, and for diurnal compo- 

 nents close to one-half. The angle v is not large at any time unless the 

 tide approaches the diurnal type. The values of sin vbja, sin vb2la, 

 etc., are therefore approxim.ately equal to bivja, b2v/a, etc., respectively, 

 and the values of cos vb^ja, cos vbz/a, etc., are nearly unity. Sub- 

 stituting these values, equation (lOA) becomes: 



Aav-\-Bibi sin Xi-{-Bibi^v/a cos a'i+52&2 sin X2-\-B2b2^vla cos X2-\- ' ' '=0 



whence: 



_ Bibi sin a'i-|-j&2&2 sin 3-2+ • • • f^^ x\ 



Aa-i-Bibi^/a cos Xi+^a^oV^- cos X2-\- • • • 



