HAKMONIC ANALYSIS AND PEEDICTION OF TIDES. 85 



an approximate representation of one of the tidal components sought. 

 These constants must, however, be modified and reduced in order to 

 be adapted to practical use. 



27. AUGMENTING FACTORS. 



In the usual summations with the primary stencils for all the 

 short period components, except component S, the hourly ordinates 

 which are summed in any single group are scattered more or less 

 uniformly over a period from one-half of a component hour before 

 to one-half of a component hour after the exact component hour 

 which the group represents. Because of this the resulting mean will 

 differ a little from the true mean ordinate that would be obtained if 

 all the ordinates included were read on the exact component hour, as 

 with component S, and the amplitude obtained will be less than the 

 true amplitude of the component. The factor necessary to take 

 account of this fact is called the augmenting factor. 



Let any component be represented by the curve 



y = A cos (at -{-a) (326) 



in which 



A = the true amplitude of the component 

 a = the speed of the component (degrees per solar hours) 

 t = variable time (expressed in solar hours) 

 a = any constant. 



The mean value of y for a group of consecutive ordinates from t/2 

 hours before to t/2 hours after any given time t, r being the number 

 of solar hours covered by the group, is 



A Ct+rl2 180 J. ~\t+rl2 



— COS {at + a) dt = sin {at + a) 



T J t~TJ2 TC ar Jt-Tl2 



dnl at + a + -^j— sin (at + a— -^j 



360 ^ , , , , . ar 360 . ar . , . \ ,„^„. 



= — COS {at + a) sm -^ = — sm -?r A cos {at + a) (327) 



Since the true value of y at any time t, is equal to A cos {at + a) 

 by (326) , it is evident that the relation of this true value to the mean 

 value (327) for the group r hours in length is 



A cos {at + a) _ irar 



360 . ar , 777T~_„ . ar (328) 



— sm ^ A cos {at -\- a) 360 sm -^ 



irar A A 



Thequantity is the augmenting factor which is to be applied 



360 sin^ 



to the mean ordinate to obtain the true ordinate. In the use of this 

 factor it is assumed that all the consecutive ordinates within the time 

 t/2 hours before to r/2 hours after the given time have been used in 

 obtaining the mean. This assumption is, of course, only approxi- 

 mately realized in the summation for any component, but the larger 

 the series of observations the more nearly to the truth it approaches. 



180 ^ . / , , ar\ 

 sm ' -^ ' 



■K ar 



