322 KEPOET— 1882. 



The undue term in the equation in which A^ has the large co- 

 efficient is 



— S 



S TfV B r^- COS [i'(r 4- i) — e] cos Xi(r + {) 



' * tan \n 



1=361 

 V 



Omitting the factor — -^^ B, sin 12n cot \n, and dropping the suffix 1 to \, 

 this becomes 



cos [(.- + \)(r + i) - £] + cos [(r - \)(r + i) - £]J 



= '-^l^tf'-^«' + '^) (' + !«=)-'] 



sin 1821(" - X) .. ,,, , ,.,-,, , 



+ — ■ — FT TT^ cos [()' - X) (- + 182) - e] 



sm jy{y — X) ^^ ^ ^ ^ -■ 



Supposing the observations to begin at noon of Januarj^ 1, ~ + 182 is 

 midnight of civil time of July 1-2, which time we may call t. 



This expression, multiplied by — q:^y It sin 12« cot \n, gives the undue 

 term due to the short tide of speed n. The maximum possible effect then 

 occurs when 



(i' + X) t - £ = 0°, or a multiple of 360° 

 („ _ X) t - £ = 0°, or a multiple of 180° 



and then the effect is 



X) sin 1821(r - X) ■) 



, ^ sin 12h f sin 182Ji(r + 



^'*" ' tani« I sin i(r + X 



h{v + X; - sin i(r - X) J 



the alternative sign being so chosen that the two terms within { } add 

 together. 



Now in the special equation which we are considering, the coefficient 

 of ^i is nearly 183, and all the other coefficients of A's and 7>'s are amall. 

 Hence an approximate value for the part of Ay which arises from the 

 influence of the short-tide n on the long-tide X, when at its maximum 

 is, without regard to sign, 



, sin 12)^ r sin 182ir'' + X) _^ sin 1821(.- - X) "1 



1S3 X 48 -R tan ^n \ sin \{i' -f X) "~ sin ^{v - X) J 



The same result arises from the discussion of the equation in which U, 

 has the large coefficient. 



There is one case in which this formula fails, and as that case actually 

 arises in the harmonic analysis of the long-period tides, it must be con- 

 sidered in detail. If either ^(i' + X) or \{r — X) is an exact multiple of 

 180° the formula becomes indeterminate. 



Suppose, for example, that r + X = 2?-7r, then 



cos[()' + X) (t + i) — {] = cos[(j' -f X) T — f] and 



364 p -| 



S C0s[(r + X)(t + i) - £] + COs[(r -\){t + i) - £] 



= 365 cos[(,' -f X) T - £] -f 5i^i^^iIll^Z^cos[(r - X)(t -^ 182) -c] 



sm i('' — X) 



"We thus see that when \{v + X) is a multiple of 180°, we must interpret 

 the indeterminate expression sin 182^(j' ± X)/sin^()' + X) as being equal 



