116 U. S. COAST AND GEODETIC SUKVEY. 



The difference between the average duration of rise and fall of the 

 tide at any place, where the tide is of the semidiurnal type, depends 

 largely upon the component M4. It is possible to obtain from the 

 high, and low waters a component with the speed of M^ which, when 

 used in the harmonic prediction of the tides, will cause the mean 

 duration of rise and fall to be the same as that at the station. The 

 effect of component M4 upon the mean duration of rise will depend 

 chiefly upon the relation of its amplitude and epoch to the amplitude 

 and epoch of the principal component Mj. By assuming an M4 com- 

 ponent with epoch such as to make the component symmetrically 

 situated in regard to the maxima and minima of the Mj component, 

 the amplitude necessary to account for the mean duration of rise of 

 the tide may be readily calculated as follows: 



Let Pi? = duration of rise of tide in hours as obtained from the 

 lunitidal intervals. 



a = Hourly speed of component Mj. =28.°984. 

 M2= Amplitude of Mj component. 

 Mg" = Epoch of M2 component. 

 M4= Amplitude of M4 component. 

 M4° = Epoch of M4 component. 



Then, for component M4 to be symmetrically situated with respect 

 to the maxima and minima of component Mj 



M4° = 2M2°±90° (446) 



in which the upper or lower sign is to be used according to whether 

 a{DR) is greater or less, respectively, than 180°. Multiples of 360° 

 may be added or rejected to obtain the result as a positive angle less 

 than 360°. 



The equations of the components Mg and M4 may be written 



2/1 = M2 cos (at + a) (447) 



2/2 = M4 cos (2af + /3) (448) 



and the resultant curve 



y^Mj cos {at + a)-\-Mi cos (2at + ^) (449) 



Values of t which will render (447) a maximum must satisfy the 

 derived equation. 



aM^sin {at + a) =0 (450) 



and for a maximum of (449) t must satisfy the derived equation 



aM^ sin {at + a)+ 2aM, sin {2at + /3) = (451) 



For a maximum of (447) 



f = ?^-^-^ (452) 



in which n is any integer. 



