42 DISCUSSION OF TIDES IN BOSTON HARBOR. 
Hence R,, and consequently the coefficient of the corresponding tidal inequality, is increased 
in the ratio of .1638 to .2342, or of 1 to 1.429, in consequence of the moon’s excess of motion at 
perigee over its mean motion. Also R;, and the corresponding tidal coefficient, are decreased in 
the ratio of .0860 to .0302, or of 1 to .851, in consequence of the moon’s decreased motion in right 
ascension when on the equator, on account of the obliquity of the ecliptic. It should be under- 
stood that the preceding ratios are not the ratios of increase or decrease compared with the whole 
tidal coefficient, but the ratio of increase or decrease of the whole tidal coefficient compared with 
the corresponding inequality having the same argument. ‘The ratio of increase of the mean tidal 
coefficient at the maximum is, in the former case, 14.0500 x 1.408=1.070, and of decrease in the 
latter, 1—.0397 x 1.408=.944. 
Since the increase of the moon’s angular motion is proportional to the increase of the parallax, 
for any increase of the parallax above the mean parallax the ratio of increase of the tidal coefficient 
of the inequality or variation is 1.429 times greater. If we also compare the moon’s motion in right 
ascension with that motion when on the equator, the increase of motion may be assumed to be as 
sivy. This is not strictly true, especially with regard to the effects depending upon the moon’s 
node, but the error, as applied to this small inequality, is insensible even in the Boston tides. For 
any decrease, therefore, of the coefficient of the disturbing force due to declination, the ratio of 
decrease of the corresponding tidal coefficient is .351. ! 
What has been stated with regard to the lunar tide is also true of the solar tide, except that 
the relative ratios of increase or of decrease are different. 
45, If we therefore put, supposing F=0, 
M=the value of A», (33), in the case of the moon, 
S=its value in the case of the sun, 
we shall have, without sensible error, 
op 3 “Cin 2ov. 
M= K(1+m < (1—n sin? v) 
(96), ; 
yn! 3 
S=e' K G +m! =) (1L—’ sin? v’) 
) 
in which the accented letters denote the same with regard to the sun which the same letters with- 
out an accent do with regard to the moon, and in which p and y must be taken at the times z and 
t; earlier. In the case of the moon we have found m=1.429, n=.351. “On account of the sun’s 
slow motion in right ascension, m/ and 2’ may be put equal to unity without sensible error. 
If we combine the separate expressions of the lunar and solar tides, as given by (33), as in the 
case of the potentials of the disturbing forces (5), we get for the combined semi-diurnal tides, 
(97) Yo= VM?+S8?+2 MS cos (y7;— a1) cos (2 ep—1—')=Q cos (2 p—I—F’) 
in which 
S sin (71—41) 
M-++458 cos (7; —.«1) 
These expressions do not include certain small corrections belonging to inequalities of a second 
order due to changes of the moon’s motion in right ascension, but these are very small and of no 
practical importance. 
(98) tan f/= 
The lunitidal interval in the preceding expression, in solar time, is 
gl . 
~ pt 
99 =1.03> 
em ; D;(2e—I—F’) 
and is equivalent to (27), putting 3; Q;=1.035 4’, and using the preceding values of P;, belonging to 
the moon only, in the rest of the expression. 
The great advantage of the preceding forms of expressions is that we dispense with all develop- 
ment in the computation of the tidal coefficient, and the trouble of taking into account a very great 
number of small terms, in the development with arguments dependi3g upon the sums and differences 
of the principal arguments, and have as arguments merely the time of the moon’s transit and the 
parallaxes and declinations of the moon and sun. The same is true also of the part of the lunitidal 
