184 Proceedings of Eoyal Society of Edinburgh. [sess. 
tion is helped, and Qw 2 takes the sign - . Thus we see that Q must 
have the sign + during the half oscillation from A to B, and the 
opposite sign — during the other half from B to A. We shall there- 
fore replace Q of the general formula by + fi, observing that the 
conclusions thence drawn only hold good from the limit v — 0 until 
the next recurrence of the same condition v = 0. This is one of the 
many instances in which generalisation by the accepted law of signs 
is inadmissible. 
Setting out from the equation 
u v= -a 2 x + fiv 2 } 
and taking the differential coefficient in regard to the primary 
variable t , we get 
2t v = - a 2 v + 2fiv lt v , 
and substituting for u v its value as above, 
2t V = - a 2 V - 2a 2 fixv 4- 2 fi 2 v 3 . 
Repeating this operation, but rejecting all terms containing fi 3 
and the higher powers, we find 
o ( V = + a 4 x - 3a 2 fiv 2 4- 2 a 4 fix 2 - 8a 2 fi 2 xv 2 
4t V = + a 4 V + 1 0a 4 fixv - 1 ia 2 fi 2 V 3 + 1 6a 4 fi 2 X 2 V 
5t V = - a?X + 1 1 a 4 fiv 2 - 10a 6 fix 2 + 84a 4 fi 2 XV 2 - 16a 4 fi 2 X* , 
and so on. 
If now we place the zero of time at the instant when the oscil- 
lating body is at the limit A where v = 0, and if we write X for the 
extreme value OA of x , the theorem of Taylor gives us 
v = 0-a 2 x^x (a 4 x + 2a 4 fix 2 ) y t -~- 
- (< vfix + 1 0a 6 fix 2 )— i + &C. , 
1 . • • 0 
and we see that only derivatives of odd orders enter into the result. 
We shall therefore save ourselves much labour in writing by tracing 
the order of formation of these odd derivatives alone. 
Let us then assume 
{2n _ 1)t V = + a 2n X - A a 2n ~ 2 fiv 2 + Ba 2 7 3x 2 - Ca 2n ~ 2 fi 2 xv 2 + Da 2n fi 2 x s , 
and operate twice thereon ; the result is 
