189 



and the surface heads may hkewise be resolved into Y components 

 whose amphtudes are H cos H° and whose initial phases are 0°; and 

 X components whose amplitudes are H sin H°, and whose initial 

 phases are 90°. 



Designating the amplitudes of the Y and X components of the 

 tide at the initial entrance as Aq cos ao and Ao sin ao respectively, the 

 amplitudes of the Y and X components of the tide at any point iiL 

 the canal are: 



A cos a=Ao cos Q!o+Siy cos H° (269) 



A sin a=Ao sin cto+Si? sin H° 



(270) 



In which ZH cos H° and 2^ sin H° are respectively the sums of the 

 amplitudes of the Y and X components of the heads in the successive 

 subsections of the canal between the initial entrance and the given 

 point. 



365. The primary tides and heads may. then be computed in terms 

 of the amplitudes of their coordinate components. After the values 

 of the components have been satisfactorily established, the amplitude 

 and phase of the resultant tide is readily determined from the 

 equations : 



tan a=A sin a/A cos a, A=A sin a/sin a=A cos a/cos a (271) 



The quadrant in which a lies is fixed by the algebraic signs of 

 A sin a and A cos a. A schedule of the values of a corresponding to 

 the value, (a), taken from a table of tangents, is set down for con- 

 venient reference. 



The primary current at a given point in the canal : 



v=B sin (a^+/3) 



similarly may be resolved into two components, one with the ampli- 

 tude of B sin /3 and the initial phase of 0°, and the other with the 

 amplitude of B cos |8, whose amplitude differs by 90° from the first. 

 366. Coordinate amplitudes of the tides for first computation. — The 

 computations are started with the tides which would be produced if 

 the instantaneous profiles were straight lines. 



