THEORY OP TIDES, 
149 
cording to the former proposition, by GS, which is to AC as 
the length of the thread corresponding to the periodical time is 
to the difference of the lengths : so that when the place of the 
body, as determined by the former proposition, without resist- 
ance, would have been S, it is actually found in G : the centre 
of attraction representing the resistance being always behind 
the body, the body will also be behind the place which it would 
have occupied without the resistance when the vibration is di- 
rect, but before it when inverted • and it will be found, that 
the forces concerned preserve their due proportion in every other 
part of the vibration. At the beginning of the true vibration, 
the body must have its greatest velocity in the subordinate vi- 
bration representing the effect of resistance, and this velocity 
must be equal and contrary to the supposed velocity in the pri- 
mitive vibration, independent of resistance ; consequently 
AC, representing the greatest velocity in the subordinate vibra- 
tion, must be equal to DI, the sine of the displacement, which 
shews the velocity in the primitive vibration. And this agree- 
i ment with the former demonstration is sufficient to show the 
accuracy of this mode of representing the operation of the 
| forces concerned. 
Theorem Q. 
If the resistance be proportional to the square of the velocity. Resistance as 
a pendulum with a vibrating point of suspension may perform jjj® y^ocit ° 
vibrations isochronous with those of the point of suspension, 
and very nearly regular, the relative situations being nearly the 
same as in the case of a similar pendulum liable to a resistance 
simply proportional to the velocity, and equal in its aggregate 
amount to the actual resistance. 
The mode of investigation which has been exemplified in the 
last scholium, may be applied to this and to all other similar 
cases j the .only difficulty being of a mathematical nature, 
since the method depends on the expression of the forces con- 
cerned in the terms of sines or cosines of arcs, and their mul- 
tiples ; and it appears to be frequently impossible to do this 
otherwise than by approximation. In the present instance we 
cannot obtaima perfectly correct expression for the square of 
the sine : the square of the sine, in the common language of 
raathe- 
