126 U. S. COAST AND GEODETIC SURVEY 
aS 
15 
The general expression for the angles of (451) and (452) may now 
be written 
at-+ (Vo-+u)—x=at-+Greenwich (Vot+u)+"2—pL—« (454) 
pL. (453) 
351. In order to avoid the necessity of applying the corrections 
for longitude and initial epoch to the Greenwich (V)+~)’s for each 
year, these corrections may be applied once for all to the «’s. 
Let 
as , 
[5 PEK —e (455) 
Then (454) may be written 
at+ (V)+u) —x«=at+ Greenwich (Vo+w)—x’ (456) 
Thus, by applying the corrections indicated in (455) to the «’s for 
any station, a modified set of epochs is obtained. These will remain 
the same year after year and permit the direct use of the tabular 
Greenwich (V,+4)’s in determining the actual constituent phases at 
the beginning of each calendar year. 
352. Let 
Greenwich (V)+u)—k’=a (457) 
then formulas (451) and (452) may be written 
h=H.+>) fH cos (at+a) (458) 
for height of tide at any time, and 
>) af A sin (at+a)=0 (459) 
for times of high and low waters. Formula (458) may be easily 
solved for any single value of ¢, but for many values of ¢ as are neces- 
sary in the predictions of the tides for a year at any station the labor 
involved by an ordinary solution would be very great. Formula 
(459) can not, in general, be solved by rigorous methods. The in- 
vention of tide-predicting machines has rendered the solution of both 
formulas a comparatively simple matter. 
TIDE-PREDICTING MACHINE 
353. The first tide-predicting machine was designed by Sir William 
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 (458). It provided for the sum- 
mation of 10 of the principal constituents, 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 constituents were afterwards 
constructed. 
