HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 105 
And therefore 
th; cos i= 4A, {cos $(1,+1,)i+cos $(1; — 1) i} 
+4B, {sin (1, +15)i—sin $(1,—la)it} + . + + 
8h, sin i= JA, {sin 3 (1, +1,)é+sin 5(1, —ly)2} 
+4B, {—cos $(1,+1,)itcos }(,—ly)i} + .. - 
Now let 
sin 3§4r 
o()=3 
o ? 
sin 5w 
so that 
il sin #§*(1, +1.) 
=) ’ 
HE) =s sin 4(1,£ly) 
We may observe that 
¢(«)=9(--2), and 9(0)=1825. 
Tf therefore % denotes summation for the 365 values from i=0 to 
i=364, we have 
VWheosl,i=lo(1, +1,) cos 182(1, + 1,) + 9(1, —1,) cos 182(1; — 12) JA» 
+[o(1, +1.) sin 182(1, +1.) —9(1, —7,) sin 182(1, —1,) |B. +. . 
Soh sin 1,i=[o(1, + ly) sin 182(1, +1.) + (1, —ly) sin 182(1, —1,) Ae 
+[—9(1,+1,) cos 182(1, +1.) +4(1, —1,) cos 182(1, — 1.) ]Bo+.. 
(68) 
In these equations there is always one pair of terms in which /, is 
identical with /,, and since ¢ (1, —/,)=1823, and cos 182 (/,—1,)=1, it 
follows that there is one term in each equation in which there is a coeffi- 
cient nearly equal to 182°5. In the cosine series it will be a coefficient 
of an A; in the sine series, of a B. 
The following are the equations (copied from the Report for 1872) 
with the coefficients inserted, as computed from these formule, or their 
equivalents :— 
