40 DISCUSSION OF TIDES IN BOSTON HARROR. 
accurately, and there are evidently some neglected sensible terms which affect this expression. 
But these angles depend upon very small quantities, since the coefficients are mostly very small, 
and consequently a very small effect throws them very much out. 
In the fourth and fifth inequalities of the heights depending upon the sun’s parallax and de- 
clination, the observed inequalities are both quite small, less than a half-inch, as theory requires ; 
but here also there are some disturbing influences not represented in the theory, for the coefficients 
have the contrary sign, and the angles of the epoch, which in this case should be sensibly 0, since 
Di, and D;7; are insensible in the expression (25), are quite large. 
These disturbing effects belong mostly to the fourth inequality having an annual period, and 
are, no doubt, due in part to the varying effects of friction, caused by annual variations in the 
velocities and positions of ocean-currents, as the Gulf Stream; for such variations, depending upon 
the changes of the seasons and of temperature, must have an annual period. We have seen that 
in the oscillations of mean level also there is a very considerable annual inequality not indicated by 
theory, which we have supposed to be due to influences of the same kind, (§ 36). Now, any amount 
of change of mean level, from whatever cause, must also produce a slight corresponding effect upon 
the range of the tidal oscillations and also upon the time, which, in very shallow seas and harbors, 
may be quite sensible to observation. According to theory, the value of B, (85) should be 0; hence 
we have an annual inequality in the times with a coefficient of four minutes, which must be due to 
the same causes, as the angle of the epoch, corresponding very nearly with that in the inequality of 
the heights, seems also to indicate. These seeming deviations from theory are merely the effects of 
slight disturbing influences not taken into account in the theory, and should be regarded as very 
important in the investigation of the subject of tidal friction in connection with ocean-currents 
varying with the seasons. 
We come now to the sixth inequality, depending upon the moon’s node. In this case U, in the 
first of (25) being sensibly nothing, we should have (1+ F) R,=P;, or substituting the value of Ps 
(18), and the observed value of Rg (85), we should have .0235 (1 +4 F)==.0375. This gives the value 
of F, as deduced from this small inequality alone, equal to nearly .6, whichis larger than the value 
before obtained, from conditions from all the principal inequalities. With the value .401 before 
obtained, the preceding equation gives Rs ——.0268, and consequently the tidal coefficient given by 
the tidal expression is too great by —(.0268— .0235) x 60 inches, or about one-fifth of an inch. If 
F—0, then the observed value of RK, should be equal to P,, the value of R, belonging to static 
equilibrium. But the difference is nearly an inch, which is entirely too great to be attributed to 
errors of observation; and hence the comparison of observation with theory, in the case of this 
small inequality alone, shows that F must have a sensible value, and that all the terms depending 
upon it which have never before been taken into account in any tidal theory, must, in the port of 
Boston at least, have very sensible values. 
The value of Bs =—2™.5 (85) in this case must depend upon EF” in the expression of Ng (27), 
since all the other terms in the expression vanish in this case. This indicates that F should have 
a positive value, as is also required in the first inequality. The effect of terms depending upon F! 
then is to cause the lunitidal intervals of larger tides to be a little greater than those of smaller 
ones, and consequently neglecting terms of a third order, to introduce small inequalities into the 
intervals proportional to the inequalities in the heights. 
All the remaining coefficients of the inequalities are quite small and unimportant, and, in the 
comparison of them with the tidal expressions as here given, there is not a very nice agreement, 
some of the residuals being nearly an inch. But the correct tidal expressions for these inequalities 
of a second order depend upon so much development that the more simple expressions, as given in 
the preceding pages, in which many terms are necessarily neglected, cannot be expected to give 
accurately these small inequalities in this case on account of the neglected terms, which, although 
in most cases insensible in the Boston tides, on account of the large value of E, must be quite sen- 
sible. The coefficients of the remaining inequalities in the times, as given by observation, agree 
very well with those given by the tidal expressions, none of the residuals being more than 0™.5. 
42. All solutions of the tidal problem, extended to the cases of different and varying motions 
of the disturbing bodies in right ascension assume, as Laplace substantially expresses it, that if 
the tidal coefficient changes with any change of the motion of the disturbing body in right ascen- 
