PROGRESS OF OUR KNOWLEDGE OF THE TIDES. 119 
available method of discussion until Sir William Thomson, 
now Lord Kelvin, developed the harmonic analysis about a 
quarter of a century ago. This analysis makes use of obser¬ 
vations made at comparatively short and uniform intervals 
of time, usually every hour, without regard to the times of 
high or low waters. As perfected by Ferrel and Darwin, it 
is now by all odds the most satisfactory form of analysis which 
has ever been applied to the tides. The tide of nature is re¬ 
garded in this method as being composed of many independ¬ 
ent waves, each one of which has a uniform speed and, for a 
given station, an invariable amplitude. Each wave is desig¬ 
nated by one or more letters, and subscripts are used to in¬ 
dicate the number of maxima in a day. The mean moon 
and the mean sun, which are fictitious bodies, give rise to 
whole groups of elements or components, among which those 
having two maxima in a day are usually very much larger 
than any of the others. Many other fictitious stars or moons 
are employed to generate waves taking account of the decli¬ 
nation, parallax, evection, and inequality of the real moon, 
and still others to represent the variations in the real sun, so 
that the aggregate is perhaps 40 or more elements or com¬ 
ponents having sensible ranges into which the natural tide 
may be broken up. Fortunately for us, however, most of 
these are so small as to be neglected without material injury, 
and we seldom work out more than about half of them. 
Until within the last few ye&vs it was impossible to com¬ 
pare the results from harmonic analysis with those obtained 
from high and low waters alone, and harmonic constants were 
only useful for comparisons between stations having similar 
analyses or for making predictions. We now have a con¬ 
venient form for computation, based upon the formulas given 
in Appendix No. 7 of the Coast and Geodetic Survey Report 
for 1894, which enables us readily to obtain the 20 following 
quantities from the harmonic constants : the mean lunitidal 
intervals for high and low waters, the four tropic intervals, 
and the tropic interval of the*diurnal wave ; the mean range, 
spring range, neap range, apogean range, perigean range, 
