HARMONIC ANALYSIS AND PEEDICTION OF TIDES. 141 



Similar formulas will represent the height difference due to the 

 other components, and the sum of all will give the resulting difference 

 in the head of water at station M and station N. 



This sum for successive values of t is readily obtained by use of 

 the tide-predicting machine, which will give the times of the maximum 

 and minimum and zero differences and also the difference in the head 

 of the water for any desired time. 



In general, the current will flow from M to N when the value of y 

 is positive and in the reverse direction when y is negative, but on 

 account of the inertia of the water there will be a lag which will cause 

 the maximum strength of flood and ebb to occur some minutes after 

 the time of greatest head, and also the slack water to be some time 

 later than the time of zero difference in head. The amount of this 

 lag may be determined from actual observations. 



In the prediction of the slack waters by the use of the predicting 

 machine the necessity of taking account of the lag for each individual 

 slack is avoided by modifying once for all the epochs determined from 

 Formula (481). 



Let to = lag or average difference between time of zero difference in 

 head and time of following slack water, and let 



t' = t + toOTt = t'-to (483) 



Then, when t represents the time of zero head, t' will represent the 

 time of the corresponding slack water. 

 Substituting in (482) we have 



y =fH cos [a^ ' + ( Fo + -m) , - (/c + aQ ] (484) 



in which t' will represent the time of slack water when y equals zero. 

 To adapt the above to the use of the Greenwich (Vo + u), we have 

 from (467) 



( 7o + m) , = Greenwich {Vo + u) +a S/15- pL, (485) 



Substituting in (484) and letting 



K'=K + ato-a Sll5+pL, = K + pL,-a (/S/15-0 (486) 



we have 



y =fH cos [at' + Greenwich {Vo + u)- k'] (487) 



In formulas (481) to (486) it will be noted that the k has for con- 

 venience been taken as referring to the longitude of station M and 

 the corresponding values for the local {Vo + u) and L are therefore 

 used. The S refers to the meridian of the standard time used in the 

 calculation. 



The k' is adapted to the meridian of Greenwich and also takes 

 account of the lag in the current. 



The elements such as represented by formula (487) may be readily 

 summed by the tide-predicting machine. While the resulting dift'er- 

 ences in head will refer to time t, the face of the machine will indicate 

 time t' , and when the difference in head registers zero, t' will indicate 

 the time of the corresponding slack water. 



Formula (487) may also be used for the prediction of the strength 

 of the current, but if the lag in the strength differs from the lag in the 

 slack waters a separate set of kq must be computed. The strength of 

 flood current will correspond to the maximum positive differences in 

 the head and the strength of ebb to the greatest negative differences. 



