HARMONIC ANALYSIS AND PREDICTION OF TIDES. 71 



stencils directly, the differences in the results obtained by the two 

 methods being negligible. For short series the irregularities are less 

 likely to be eliminated, and since the labor of summing for such a 

 series is relatively small, the abbreviated form of summing is not 

 recommended. As the length of series increases the saving in labor 

 by the use of the secondary stencils increases, while the irregularities 

 due to the short process tend to disappear. It is believed that the 

 use of the secondary stencils will be found advantageous for all series 

 more than six months in length. 



In the primary summations there are obtained 24 sums for each 

 page of tabulations, representing the 24 component hours of a com- 

 ponent day. In general each sum will include 7 hourly heights, and 

 the average interval between the first and last heights will be 6 

 component days. A few of the sums may, however, include a greater 

 or less number of hourly heights within limits which may be a day 

 greater or less than 6 component days. 



Let the component for whidi summations have been made by use 

 of the primary stencils be designated as component A and the com- 

 ponent which is to be obtained by use of the secondary stencils as 

 component B. For convenience let it be first assumed that the 

 heights included in the sums for component A refer to the exact 

 component A hours. This assumption is true for coniponent S but 

 only approximately true for the other components. It is now pro- 

 posed to assign each hourly page sum obtamed for component A to 

 the integral component B hour mth which it most nearly coincides. 

 Component A and component B hours separate at a uniform rate, and 

 the proposed assignment will depend upon the relation of the hours 

 on the middle day of each page of tabulations.. The tabulated hourly 

 heights on each full page of record run from zero (0) solar hour on the 

 first day to the 23d solar hour on the seventh or last day of the page. 

 The middle of the record on each such page is therefore at 11.5 solar 

 hour on the fourth day, or 83.5 solar hours from the beginning of the 

 page of record. 



Let a and & represent the hourly speeds of the components A and B, 

 respectively, and p and p^ their respective subscripts, and let n equal 

 the number of the page of tabulation under consideration, beginning 

 with number one as the first page. 



The middle of page n will then be 



[168 {n-l)+83.5] or (168 n- 84.5) solar hours (265) 



from the beginning of the series. 



Since one solar hour equals a/15p component A hours (formula 

 261), the middle of page n will also correspond to 



(168 n — 84.5) :j-^ component A hours (266) 



from the beginning of the series. 



As there are 24 component hours in each component day, the 

 middle component A day of each page will commence 12 component 

 A hours earlier than the time represented by the middle of the page, 

 or at 



[(168 n-84.5) ^ — 12] component A hours (267) 



from the beginning of the series. 



