152 
THEORY OF TIDES. 
Failure of 
direct investi- 
gation. 
Simple pendu- 
lum with resis- 
tance as the 
square of the 
velocity. 
*0022, shews the true extent of the excursion *9178 ; and reck- 
oning from this point as the beginning, the displacement in the 
middle will be reduced to about f 05. Now an equal mean re- 
sistance, varying simply in proportion to the velocity, would 
cause a displacement in the middle of '3957 instead of *4434 j 
and reckoning from the true beginning of the vibration, the dis- 
placement in the middle would vanish, instead of being reduced 
to about ‘05, and the extent would be 9196 instead of '9178. 
And if r were smaller than there would obviously be still 
less difference in the two cases. From the small proportion 
which the second displacement bears in this case to the first, it 
may be inferred, that any further calculation, of the effects of the 
third order, would be wholly superfluous. 
Scholium 1. Dr. T, has suggested an ingenious method, 
which affords a formula for the coefficients of the first series $ 
but unfortunately it loses its convergence too soon to be of any 
use. Taking the equation — aqr+3lq3x+5cq5x+...= 
2 (1 — <fjc) “%r, we may expand this expression, by means of 
the binomial theorem, into 2qv — § 3 .r — igl * — and 
substituting the cosines of multiple arcs for the powers of qx, 
and then comparing the homologous terms, we obtain a= 2 
(.1 — a— 0— y— ...) where *==-1-773 0 = i-i-i a, y=4.- ■§.•£& 
^ — T-i • -fy, V-* ••• ■ £=—4 (a+0 + y + ...) where 
£=4.4. %dy y = 44.1/3, £ = 44.| y : and c = — ■ f 
(«+£+y+. •) where a 0 = ft • 
but in all these series it is obvious that the ratio of the terms, 
as they diminish, approaches to equality ; so that it is even diffi- 
cult to determine whether or no this sum is finite. But a still 
greater objection to this method is, that the third fluxion 
— treated in the same way, affords a very different 
result. 
Scholium 2. In the case of a simple pendulum, subjected for 
a single vibration to a resistance proportional to the square of 
the velocity, the space described may be correctly calculated by 
means of a logarithmic equation, and the time might also be ex- 
pressed, if it were required, in a series. Let the space described 
be x, and the resistance^, then the force may be represented by 
1 — x — y, and the square of the velocity will be j (4 — xx — yx), 
whence y^=cL J (#— &•#— yx), and y-~-ax — axx~~-c. yx , and y -\-ayx 
