J56 
THEORY OF TIDES. 
Determina- 
tion of the 
greatest ex- 
cursion. 
* and c $ for b must always be 1 -f2‘9 2 ==9'4l* in order that 
the equation may hold good for the greatest and least vibrations : 
consequently a-fc=5*8. We may first allow Sl±i to a , in 
order to form with b a result similar to the true compound 
vibration, and the remainder 2*663 must again be distributed 
between a and c in such a proportion that the interval of the 
greatest vibrations may be 44 360°, and m must be so determined 
for this purpose, that ^—*-§4 shall be equal to -14, whence 
m=4-, and for this part of a, 'a : c= 1 : -f-, 'a ; a-fc=l : f , 
and *-.2*663=1 *664, a= 4 80, and c— 1, so that the 
velocity hecomes 4 , 8g.r49*4l94|.r+ < ;44 r : and f° r the interval of 
the least vibrations, — +1 -.34= and the whole is found 
* m+ 3 o 2 s 1 9 * 
1 ttst 
3*85J , +9 , 4l5-|-J»+29||.^and for a mean valueof the coefficients. 
4*3 ? T+9*4l 9 f |a ?+ 1 *5?- |f a?. 
If now we denote the ratio of the square of the time of the 
most frequent vibration © to that of the square of the time of 
the spontaneous vibration of the pendulum ($2) by the ratio of 
n to X, the corresponding displacement will be to the distance 
expressive of the force as -4L to 1, and the term 4*8^ will 
exhibit a displacement of : and in the other terms, sub- 
re— 1 
stituling, for and (44) respectively, we have l- 0 - 7 - 
1*14 5re re j re - .... , 
and 77-7: or — ■ — ? and for multipliers: and 
1 14ow — 1 n— *9.344 re — *871 r 7 
the distance thus determined shows the place in which the 
body would have been if there had been no resistance, which is 
before the true place when the multiplier is positive, and behind 
when it is negative : the distance of this virtual place there- 
fore becomes from £r+3£fAr, Sr + 3:Sf .■§* +rn + —^7 
9344 
541 x)> and the velocity 9 *+2*9 ? -§-§. a ?— rn 
(Si s *+ « »*)• Here !t is 
obvious that as n approaches to 1, to *0344 or to "S/l, the value 
of the corresponding term increases without limit, and the 
period of the resistance may approach to that of the slower 
vibration, or may even exceed it, in very particular circum- 
stances : 
