106 



U. S. COAST AN"D GEODETIC SUEVEY. 



to some extent from that just described. In this discussion it is 

 assumed that a series of 365 days is used. 



Let the entire tide due to the five long-period components already 

 named be represented by the equation 



y = A cos {at + a) +B cos (bt + /S) + C cos (at + 7) 

 + D cos {dt + 8) +E cos (et + e) 



(424) 



For convenience in this discussion let t be reckoned from the 11.5th 

 solar hour of the first day of series instead of the midnight beginning 

 that day. Every value of t to which the daily means refer ^viU then 

 be either or a multiple of 24. 



Let A', B', C, D', and E', equal 



A cos a, B cos )3, C cos 7, D cos 5, and E cos e, respectively, and 

 A", B", C", D", and E" , equal 

 — A sin a, —B sin /3, — C sin y, —D sin 5. and — E sin e, respectivelv. 



(425) 

 Then formula (424) may be written 



y =A' cos at + B' cos M-r C cos ct-rB' cos dt-\- E' cos et 

 + A" sin at + B'' sin U+C" sin d + D" sin dt + E" sin et (426) 



In the above equation there are 10 unknown quantities, A', A", 

 B' , B" , etc., for which values are sought in order to obtain from them 

 the amplitudes and epochs of the five long-period components. The 

 most probable values of these quantities may be found by the least 

 square adjustment. 



Let ?/i, yn, .... 2/365 represent the daily means for a 365 day series, 

 as obtained from observations. If we let n be any day of the series, 

 the value of t to which that mean applies will be 24(n— 1). By 

 substituting in formula (426) the successive values of y and the values 

 of t to which they correspond, 365 observational equations are formed 

 as follows: 



y^ = A' cos 0+5' cos 0-r .... 

 + A" sin Q + B" sin 0+ .... 

 y. = A' cos 24a 4-5' cos 246+ .... 

 '+^" sin 24a + 5" sin 24&+ .... !^(427) 



?/365 = ^' cos 24x364a + 5' cos 24X364&+ .... 



+ ^" sin 24x364a + 5" sin24x364&+ .... ^ 



A normal equation is now formed for each unknown quantity by 

 multiplying each observational equation by the coefficient of the 

 unknown quantity in that equation and adding the results. Thus, 

 for the unknown quantity A' , we have 



2/1 cos = ^' cos^ + 5' cos cos 0+ .... 



+ ^" sin cos + 5" sin cos 0+ .... 

 2/2 cos 24a = ^' cos^ 24a. + 5' cos 246 cos 24a + . • • • 



+ A" sin 24a cos 24a + 5" sin 246 cos 24a + • • • 



2/365 cos (24 X 364a) ^A' cos" (24 x 364a) 



+ 5' cos (24x3646) cos (24 x 364a) + 



+ ^"sin (24x364a) cos (24 x 364a) 

 + 5" sin (24 x 3646) cos (24 x 364a) + 



> (428) 



