154 THEORY OF TIDES. 
corrected displacement : and this correction will, in a similar 
& I 
manner, afford a second of "Z2 a* 6 j so that the true displace- 
ment becomes \ a* 2 — T^a* 4 + ci xG — TJ ax * — ...which, a s 
is well known, is equal to a— «<;* : and the diminution of the 
velocity ax — TT 3 ax 5 + •— a* 5 — .. — aSi ; which will, of 
course, vanish, when = iso° $ so that the body will be at 
rest at the expiration of the corresponding time of a complete 
vibration in one direction. And a similar mode of calculation 
may be applied to the case of a simple pendulum, with a resis- 
tance varying as the square of the velocity, except that here the 
variation of the resistance at each step makes the process more 
complicated. 
Theorem D. 
Peculiarities If the resistance be proportional to the square of the velocity, 
vibrabo^Twhli a P eR dnlum, °f which the point of suspension performs vibra- 
resistance. tions composed of two regular vibrations, may have its greatest 
excursions a little after the greatest excursions of the point of 
suspension when its vibrations are inverted, and a little before 
them when they are direct, provided that the slower vibrations 
be the larger. 
In order to express the resistance as correctly as possible in 
this case by a series of multiple arcs, it would be necessary to 
have a great variety of terms, some approaching in their periods 
to the primitive vibrations, others triple and quintuple of these : 
but for the present purpose these greater multiples may be safely 
omitted, taking care only that the omission do not affect the 
determination of the coefficients of >he rest. The general 
methods of obtaining a series in the terms of sines and cosines 
of multiple arcs fail here, as before, on account of the positive 
terms resulting from the squares of negative quantities, where 
the conditions of the problem require that they should be 
negative, and it is necessary to employ approximations obtained 
from the results of individual substitutions. For this purpose a 
series of five or six terms has been tried in various ways without 
success : and the most convenient form which has been discover- 
ed consists of three only, two isochronous with the primitive 
vibrations. 
