THEORY OF TIDES, 
151 
the supposed centre of attraction for the point of suspension, 
and the mean place for the vertical line : but in the following 
terms, the value cf n is successively reduced to 
and so forth : consequently, the whole displacement immedi- 
tely produced by the effect of the resistance rS~x will be rn 
Sx - : 1696 S3* - -° 244 ^ S 7* - ^ 
\n — i n — 9 n — 25 n — 49 n—<Sl 
S 9 X — ~~~ S ilx). This displacement will, howevet, 
cause an alteration of the resistance, which may be considered 
as a differential of the former, and since ( y <2 )‘ =2 yy, the new 
resistance mav be expressed by the product of the new and 
twice the original velocity, or by — 2 r q nsx cx — ^ 
<l3x — • 5 and the consequent displacement 
may be determined in the same manner as for the original re- 
sistance. The first term gives 2rV* S2x : for the re- 
mainder we must find an equivalent series in terms of the ’co- 
sines of multiple arcs, since the direction of the force of resist- 
ance does not change where the sine becomes negative: and 
each term will require a separate investigation while n remains 
indeterminate j but for the present purpose two of the terms 
will be sufficient. The method employed by Euler for deter- 
mining the coefficients in such cases is of no use here, because 
it affords a progression of sines only : we must, therefore, put 
a(ix+b<;3x+cc,5x-\- dqjx successively equal to SxgSx, and to S- X f5r ; 
then, bisecting and trisecting the quadrant, we find the coeffi- 
cients 1667, — *3333, *6869, and— - -5202, and — *1057, 
— '2880, *0421, and — '2478, respectively, and the whole of 
this second displacement becomes 2r 2 « 2 S L 2x -~ 
r __ 1,5202 - ' '105 7 
cs*+ 2552. 
n — I ^ n — 9 ’ n— 25 
5088 
9 
9* 
'2886 1 
$JX + 
n — 9 ’ 
•6421 
«— 25 
•24.78 
c :tx CV x 
1—49 5 
>)• 
For example, if we take r and n, each equal to the first Example, 
formula gives — -‘4434 for the displacement at the middle of the 
time, and 0 for the beginning - } the second 0 at the middle, and 
—•0022 at the beginning : but the true beginning of the actual 
vibration is modified by the velocity belonging to the first order 
of the effects of resistance, which is found in this case '390, 
consequently, the true time of rest will be when the velocity is 
— *390 in the primitive vibration, or when the arc corresponding 
to the excursion is 67°, and its sine ‘920, which, lessened by 
* 0022 , 
4 
