THE LATITUDE-VARIATION TIDE. 
Ill 
sin v j8 r sin N m v ft r 
sin ft sin (v ft — p ^ sin (v ft + P 
w _ sin v ft, sin N to > ft ,_ ( 
ft, sin (v ft, - p » (* ft, + p 
and includes neither y nor i. The summations with re¬ 
spect to k are from k = 0 to k — m — 1. Equations (8) take 
the place of the last two series of equations given by (6), and 
afford by their solution the c’s and s’s, and hence the a’s and 
s’s, for substitution in the forms (4). 
The generality and flexibility of this solution may be re¬ 
marked. The ordinary solution, when the period of the 
inequality is known in advance and the observations are 
grouped in accordance with that period, is obtained from it 
by putting i —j. The method here given applies whether 
the period of the inequality sought is known in advance 
exactly, with considerable precision, or only roughly, and I 
would suggest that it might prove useful in picking up ine¬ 
qualities when nothing is known of their periods. The 
transference of the secondary time zero to the midde instant 
of the observations preserves in both the solutions of this 
paper the inherent symmetry and simplicity of the forms • 
in the last one it effects a complete axial revolution, separat¬ 
ing the cosine from the sine coefficients, and thus notably 
diminishing the labor of solution. From a theoretical point 
of view, the most remarkable thing about the second solution 
is that the/-coefficients, to which is assigned the title role in 
the formation of the normal equations, are not the quantities 
sought, and when the solution is effected it inures to the 
benefit of other coefficients—that is to say, other coefficients 
are thereby determined. I do not recall any other instance 
in mathematics where a like distinctively vicarious action 
appears or is noted. That this vicarious solution is logically 
sound may be shown in various ways. It may be made to 
repose upon the fact that the certain and only possible value 
10—Bull. Phil. Soc., Wash., Vol. 13. 
