THEORY OE TIDES. 
155 
vibrations, and the third having a recurrence less frequent by 
one time in the common period than the slowest of these : then 
the coincidence being established at the time of the greatest and 
least excursions, and at the transit of the vertical line nearest to 
the middle of the intermediate time, a mean value of the 
coefficients may be obtained, which no where differs very 
materially from the truth ; although, if we desire to make the 
coincidence more perfect in any given part of the period, we 
may do it by altering the values of the coefficients a little; and 
by these means we may obtain a correction of the approxima- 
tion, sufficiently near to the truth We may also suppose the 
actual compound vibrations to preserve their regularity without 
any material deviation, following the same law as if the resist- 
ance were either inconsiderable, or varied simply as the velocity; 
and we may make the proportion of the greatest to the least 
actual vibration that of ?/z + l tom-1 ; then calling the periodi- 
cal time of the greater primitive vibration ( D ) t, that of the 
lesser (o) being unity, and x being the rc corresponding to the 
time in the latter, beginning with the perfect coincidence in the 
vertical line, the distance from that line at any subsequent 
time will be expressed by 5^ + m 5 y ; and the velocity by 
+ creating a resistance which may be called 
r which has already produced a displacement 
determinable as in the former proposition, whence we may 
obtain from the true place the place in which the body would 
have been found if there had been no resistance. In order to 
facilitate the computation, we may assume particular values 
of in and t, making the one 3, and the other 1^- ; and then 
determine the coefficients of the formula a^ x -\-b^-lx+c'^ x =z 
so as to obtain as correct a coincidence as possi- 
ble of the magnitude and of the period of the joint vibrations 
at the time more immediately to be considered. Now it is 
easily shown, from the well known properties of compound 
vibrations, as applied to the intervals of successive spring and 
neap tides, that the interval between two of the greatest vibra- 
tions will be expressed very nearly by 300°, and the inter- 
val between two of the smallest by ——t 360°, provided that 
m—t 
the periods differ but little from each other : and from these 
formulas we must determine the proportions of the coefficients 
a and 
