122 U. S. COAST AISTD GEODETIC SURVEY, 



Lot 



(^^15-pL-K^=-K' (469) 



Then (468) may be written 



at + ( Fo + u)-K = at + Greenwich ( Fo ^ u) - k ' (470) 



Thus, by applying the corrections indicated in (469) to the k's for 



any station, a modified set of epoclis is obtained. These will remain 



the same year after year and permit the direct use of the tabular 

 Greenwich (Fo-r'u)'s in determining the actual component phases at 

 the beginning of each calendar year. 



Let 



Greenwich ( Y^ + u) -k' = a (471) 



then formulas (465) and (466) may be written 



h = H^ + :Z:fn cos {at + a) (472) 



for height of tide at any time, and 



S a/H sin (at + a) = (473) 



for times of high and low waters. 



Formula (472) may be easily solved for any single value of t, but 

 for many values of t as are necessary in the predictions of the tides 

 for a year at any station the labor involved by an ordinary solution 

 would be very great. Formula (473) can not, in general, be solved 

 by rigorous methods. 



The invention of tide-predicting machines has rendered the solu- 

 tion of both formulas a comparatively simple matter. 



The first tide-predicting machine was designed by Sir WUliam 

 Thomson (afterwards Lord Kelvin) and was made in 1873 under the 

 auspices of the British Association for the Advancement of Science. 

 This was an integrating machine designed to compute the height of 

 the tide in accordance with formula (472). It provided for the sum- 

 mation of 10 of the principal components, and the resulting pre- 

 dicted heights were registered by a curve automatically traced by 

 the machine. This machine is described in Part I of Thomson and 

 Tait's Natural Philosophy, edition of 1879. Several other tide- 

 predicting machines designed upon the same general principles, but 

 providing for an increased number of components were afterwards 

 constructed. 



The first tide-predicting machine used in the United States was 

 designed by William Ferrel, of the U. S. Coast and Geodetic Survey. 

 This machine, which was completed in 1882, was based upon modified 

 formulas and differed somewhat in design from any other machine 

 that has ever been constructed. No curve was traced, but both the 

 times and heights of the high and low waters were indicated directly 

 by scales on the machine. The intermediate heights of the tide could 

 be obtained only indirectly. A description of this machine is given 

 in the report of the U. S. Coast and Geodetic Survey for the year 1883. 



The first machine designed to solve simultaneously formulas (472) 

 and (473) is the U. S. Coast and Geodetic Survey tide-predicting 



