118 TJ. S, COAST AISTD GEODETIC SUEVEY. 



Substituting the above in (458) we have 



M,_ , sin (90°-^ai9i?) _ , cos UDR , . 



M;"^' cos (180° -aZ>i2)~"^ 2 cos aDR ^^^^^ 



and 



M4 = T i Ht^ M' (462) 



^ cos aL'ic 



M4 must be positive, and the sign of the above coefficient will 

 depend upon whether aDR is less or greater than 180°. 



The approximate constants for components S,, N,, Kj, and O^ 

 may be obtained from the observed high and low waters as follows: 



Add to each low-water height the mean range of tide. Copy the 

 high and modified low water heights into the form for hourly heights 

 (Form 362), always putting the values upon the nearest solar hour. 

 Sum for the desired components, using the same stencils as are used 

 for the regular analysis of the hourly heights. Account should be 

 taken of the number of items entering into each sum and the mean 

 for each component hour obtained. The 24 hourly means for each 

 component are then to be analyzed in the usual manner. 



The results obtained by this process are, of course, not as depend- 

 able as those obtained from a continuous record of hourly heights. 

 The approximate results first obtained can, however, be improved by 

 the following treatment if a tide-computing machine is available. 



Using the approximate constants as determined above for the 

 principal components and inferred values for smaller components, 

 set the machine for the beginning of the period of observations and 

 find the predicted heights corresponding to the observed times of the 

 high and low waters. Tabulate the differences between the observed 

 and predicted heights for these times , using the hourly height form 

 and entering the values according to the nearest solar hour. These 

 differences are then to be summed and analyzed the same as the 

 original observed heights. In this analysis of the residuals the com- 

 ponent Mo should be included. The results from the analysis of the 

 residuals are then combined with the constants used for the setting 

 of the predicting machme. 



In making the combinations the following formulas may be used: 



Let A' and k' represent the first approximate values of the constants 

 of any component. 



A" and «", the constants as obtained from the residuals. 

 A and k, the resultant constants sought. 



Then 



A=-^{A' cosk'+A" cosK"y+{A' sin «' +^" sin «")' (463) 

 and 



^ A' sin k' + A" sin k" ,,^,. 



K = tan 1 -n > , a„ Tf (464) 



A cosK +A cosK 



33. HARMONIC PREDICTION OF TIDES. 



The methods for the prediction of the tides may be classified as 

 harmonic and nonharmonic. By the harmonic methods the ele- 

 mentary component tides, represented by harmonic constants, are 

 combined into a composite tide. By the nonharmonic methods the 



