Q94 REPORTS ON THE STATE OF SCIENCE, ETC. 
PART II. 
Methods. 
§ 10. Inference of chief semi-diurnal constituents.—It is unnecessary to enter 
into the details of this process. A tidal constituent being expressed as R cos (ot—e), 
where ¢ is zero at midnight on January 0-1, 1918, the following values of R and 
—e were actually used :—- 
R —e€ 
ise g 1:870 180:00° 
M,: 5'633 150°33° 
1 eS 1-100 12°53° 
K, : 0-600 37:46° 
1 ae 0-300 105°33° 
An analysis for 8, was carried out using six months’ observations, and the 
average values of ratios of amplitudes and differences in lags for all other con- 
stituents (referred to S,) were fourd from known harmonic constants for ports 
within a few hundred miles of Newlyn; from these were deduced the values of R 
and —e given above. 
Regarded as true representatives of the constituents, R is probably correct to 
the nearest tenth of a foot and « may be in error by several degrees. The figures 
represent precisely what was removed by accurate arithmetical processes. 
§ 11. Calculation of harmonic constituents.—The process on which the intensive 
analysis of tidal observations depends is that of the summation of harmonic 
constituents, Consider the calculation of the five constituents given in the 
previous paragraph; the calculation of the individual arguments by successive 
addition of hourly increments is itself no light task, and the consequent deter- 
mination of the cosines and the multiplication by the appropriate amplitudes is a 
task of appalling magnitude. The greater part of this labour has been avoided in 
the scheme now to be explained. 
Any term R cos (ot—€) can be written in the form 
D cos (ot—e+d) + D cos (ot—e—d) 
where 2D cos d=R, and D can be chosen at will. By choosing D=10,1,01,... 
we thus avoid the labour of multiplication, but double the labour of determining 
arguments and cosines. The latter, however, can also be avoided by the construc- 
tion of a suitable abac. 
Let the argument at time ¢ =o for a given harmonic term be a, and let the speed 
be a, so that we have to construct cos (ot + a) at unit intervals of ¢t. Suppose that 
we have a horizontal scale graduated uniformly in degrees (@) on one side, and on 
the other side let there be the appropriate cosine scale, graduated, say, at intervals 
of 0°01 in cos 6. Then if we mark on the scale the values @=a,a+0,a+20,.... 
a+to,.... wecan atonce read off, by interpolation in the cosine scale, the values 
of cos a, cos (a+), ... . to three decimals. This double scale avoids reference to 
trigonometrical tables or to a graph of cos @. 
But this method does not perform automatically the processes of adding ot to 
the argument a for the required values of ¢; moreover, a very lengthy scale would 
be required for us to be able to read off many values to the required degree of 
accuracy. The problem can be solved by cutting up the scale into sections of 
length 6=0; the sections @=0 to o, o to 2c, .. . . are then placed parallel to one 
another vertically with their extremities on horizontal lines. A large number of 
sections can thus be drawn on an open scale and placed side by side. Suppose that 
the given value of the initial argument (a) be in the first section ; then a horizontal 
straight line passing through this point will cut the vertical sections in the points 
corresponding to @=a,a+o,....a+ot,.... andby interpolation in the vertical 
cosine scales the values of the cosines can be immediately read off. It is obvious 
that only one angle (a) needs to be determined, so that it is not necessary to have 
the 6-scale marked on the cosine scale at all. The best procedure is to use paper 
ruled in quarter-inch squares with the vertical lines half an inch apart, and with 
one half-inch to a degree. Since we know that the vertical sections have their 
upper extremities corresponding to 9=0, 0, 20,... ., then any intermediate angle 
