186 
Proceedings of Boy al Society of Edinburgh. [sess. 
2ai t ~at 
1 + 2 ri 
-g-a/3X 2 
1 
2 at (2 a<) 3 
T + 1.2.3 
+ 2 
T 
1.2.3 
+ 2 
(2aQ 5 
1...5 
(aQ 5 
1...5 
+ &c. 
- &c. 
or 
^ a^X 2 {2sin(aO - sm(2a t)} . 
The coefficients D succeed each other in a very complex manner ; 
I have not succeeded in separating them into series of powers, the 
nearest approach giving the formula 
2 
15 
{5 sin (at) - 4 sin (2 at) + sin (3a£)} 
and thus we must be content to represent v by the formula 
v= -aX. sina/+ i a/3X 2 {2sina£-sin2a£} 
o 
+ ay3 2 X3 | 
16^ + 216 ^--2232^ + &c 
...5 ...7 ...y 
and, for the reasons already given, we may neglect this term con- 
taining y8 2 . 
Now when at — 7r, its sine is zero, but so also is the sine of 
2a t = 27r, wherefore v is zero when t = — ; in other words, the air’s 
resistance does not influence the time of oscillation. 
The value of x may be found in the same way, observing that 
the first derivative of v is the second derivative of x, and so on ; or 
it may be found by integration from the value of v. Either process 
x = X cos at + 
which when a£ = 180° gives 
6 COS 2 at - — cosa t | 
X' = -X + ^-/3X 2 , 
that is to say, the length of the oscillation is shortened by-g-/3X 2 . 
Thus we see that, while the air’s resistance lessens the extent of 
the oscillation, it also lessens the velocity, and in such a way that 
the body is brought to rest in the same time as if there had been 
