THEORY OF TIDES. \t 
= (a — ax) x. Then if we mak ey=uz, % being a function of 
x > and j/=a3+2io we have uz-\-zu-\-auzx— (a — ax)x 3 and in 
order to determine z, we may put uz+au%x =o,-§ +ax—o, 
HL«= —ax, or zz=e— ax , and, substituting for Z u, e~~ a *u = 
(a— ax) x , or, if ai—w, e~ w u—w — ~ a w, u=e w iv—f e^w w 3 
an d y=er^Je^%v—^e~ , ~ w Je^’wzv. Hence (Eul. calc, integr. pr. 
52)y=Ce-™+\ — ~Be~^—~w4-j;—Ce- ax + 1 — 
which must vanish when x=o , and e~ ax —l, whence C — i* 
«-H a+1 
D==— — and y= - (1 — e~~ ax ) 
\ v 
Scholium 3. The same result may be still more simply ob- Different in- 
tained, by repeated approximations, from the first expression vesti S aUoil ‘ 
y—a f ( x — x'x — yx) : neglecting first the term yx, we have 
y=ax — \a,x> then substituting this value in yx, we have afylt 
= \ a 2 * 2 — a** 3 , and y—a x — \ ax 2 — ^ x 2 + ^ a 2 * 3 , to 
which we add, by another similar operation, the terms +1^1 
flV — ZT* a * xi > an< ^ whole becomes ax — |a 9 * 2 -}- a 5 * 3 — « 
— 4 a * S +Txl a 5 * 3 — ••• But it is well known that 
c~ ax = 1 — a*-f \a?x 2 '— ^~a 3 * 3 ..., therefore, l—er~**—'her«- 1 - 
-f — x ; precisely as before. The length of the whole vibra- 
tion may obviously be found by making y=0 : and if it be de- 
sired to determine the time, we must develope, by means of the 
polynomial theorem, the expression + 
(a 3 .}. a 3 )* 3 — ...) *—£, and take the fluent. 
Scholium 4. This mode of exhaustion may be illustrated by Pendulum 
another well-known case of a pendulum with a constant resis- 
tance, which is known not to alter the time of vibration. Calling 
the initial force 1, and the resistance a , the velocity destroyed 
will be ax, if x be tbe arc proportional to the time : and the 
diminution of the space will be ^ax q : but this displacement 
will cause a new force, which is to the initial force 1 as 4 axt 2 to 
the radius 1, and is therefore represented by \a x ^, hence the 
velocity will be increased by the quantity J\a x<i x ==ixT ''* 3 > all d 
the space by a ** : so that we have 4^ 2 — ^ a* 4 for the 
corrected 
