12 DISCUSSION OF TIDES IN BOSTON HARBOR. 
tions of condition for determining A, and 4 for m values of ¢ =47;, When A, and A; and also 4’ have - 
been determined from preceding conditions. 
With m values of H,, also, for both high and low waters of both transits, (52) gives, when A; 
and 4’ are known or insensible, m equations of condition for determining A, and 4 fo m values of 
g from observations of the heights only. 
From (32) we have 
(53) A,=K, sin (g — z)=M sin 9+ N cose 
in which 
(54) K,=V M+? and tan —— 
7 J 
With m values of A;, determined by the preceding equations, (53) gives m equations of condi- 
tion for determining R and «. , 
From the first of (41) we obtain, approximately, when A, is small in comparison with- A2, and 
consequently sin q is small, 
and from the second of (41), 
__ Arcos 4 cos 4 
AAS en 
When @ and @—3= are very small we can use for A, its mean value K,, and then with the 
preceding value of A, we get 
(2 
Ky sin 4 
ae ( i 2K, sin (g—a) 
i) 
; ) jy = SOS I (g—2) 
4K 
The values of q; and q, ave the complements of the preceding expressions respectively. 
TABLES OF AVERAGE OR NORMAL VALUES. 
26. The following tables contain the averages of groups of observations taken within certain 
limits of two arguments, and arranged, with reference to the averages of the arguments to which 
the observations correspond, in the form of tables of double entry. By summing these average 
results in two ways, the averages of all the observations contained within the limits of each group 
of either argument are obtained. The arguments have been for the most part divided into twenty- 
four equal parts, and the mean of the two limits of each division has been taken as the average of 
the values of the arguments, except in the case of (y—vw’), in which the true average has been 
obtained. From these tabular results all the constants in this discussion have been obtained, and’ 
they might be treated in various other ways and many important results obtained which have not 
been brought out here. ; 
An explanation of the notation contained in the headings of these tables may be found in sece- 
tions (4), (8), and (11). The values of all the arguments except 7 are given for the time of the 
moon’s apparent transit over the Washington meridian, happening a little more than two days 
before the time of high water, and which is the transit C according to Lubbock’s notation. For the 
sake of convenience in grouping the observations, the values of 7 were used for a time two lunar 
days later, for which a reduction must be made when it is necessary to have the values of the argu- 
ments all referred to the same time. 
To these tabular values a constant of 2 days must be added to the lunitidal intervals, and also 20 - 
feet to the heights of high water, and 10 feet to those of low water, for the absolute heights above 
the assumed zero of the tide-gauge. 
