88 REPORT—1883. 
For the purposes of using the forms for harmonic analysis of the tidal 
observations, these formule may be reduced to more convenient and 
simpler forms. 
The mean values of N and p, are required, and for the treatment of 
the L and M, tides the mean value of p—é, denoted by P. For deter- 
mining these three quantities, we may therefore add half the coefficient of 
T once for all, and write 
N=276°-2861 —0°-05295 D—19°34146 T 
py=280°-8833 +.0°-00005 D+ O-ol71 TE: - - : OY) 
P+f=261°0 +40°111D 440°69 7 
where 7’ is simply the number of years, whether there be leap-years or 
not amongst them, since 1880, and D the number of days from Jan. 1, 
numbered as zero up to the first day of the year to be analysed. 
Now, suppose d to denote the number of quarter days either one, two, 
or three in excess of the Julian years which have elapsed since 0% Jan. 1, 
1880, up to 0» Jan. 1 of the year in question; let D denote the same 
as before; and let LZ be the East Longitude of the place of observation in 
hours and decimals of hours. 
Then for s,, p,, /., the values of s, p, h at 0 of the first day, we have 
85=150°'0419 + 132°-67900 T+ 3°'29410 d+13° 1764 D—0°'54902 “1 
Po=240°'6322+4+ 40°-69035 7'+ 0°-02785 d+ 0°-1114 D—0°-00464 Z | (60) 
h,=280°5287+  0°:00769 T+ 0°24641 d+ 0°-9856 D—0°-04107 1 
In these formule 7’ is an integer, being the excess of the year in ques- 
tion above 1880, and d is to be determined thus:—if the excess of the 
year above 1880 divided by 4 leaves remainder 3, d is 1; if remainder 2, 
it is 2; if remainder 1, it is 8; and if remainder zero, it is zero. For 
example for 1895, T=15, d=1; because from 0 Jan. 1, 1880 to 0» Jan. 1, 
1895, is 15 Julian years and a quarter day. For all dates after Feb. 28, 
1900, one day’s motion must be subtracted from s,, p,, h,, p;,; P+é, and 
one day’s motion added to N. 
The terms in L may be described as corrections for longitude. 
The 13 x 360° and 860° which occurred in the previous formule for 
s and h are now omitted, because 7’ is essentially an integer. 
If it be preferred, the values of s, and N may be extracted from 
the Nautical Almanac, and h, is (neglecting nutation) the sidereal time 
reduced to angle. We may take p, from a formula given by Hansen at 
p. 300 of the Tables de la Lune. This latter course is that which is 
followed in the forms for computation. 
$7. Summary of Initial Arguments and Factors of Reduction. 
Tue results for the various kinds of tide are scattered in various parts 
of the above, and it will therefore be convenient to collect them together. 
In order to present the results in a form convenient for computation, 
each argument is given by reference to any previous argument which con- 
tains the same element. In the following schedule Arg. M, and Fac. M, 
(for example) mean the argument and factor computed for the tide My, 
