292 Proceedings of Boy cd Society of Edinburgh, [june 20 , 
Hence, after having computed the fourth crossing it was easy for 
us to see that the fifth must he between 55 and 56 ; and now that 
the sixth crossing has been accurately determined we readily infer 
that the seventh must he at 112*48, the eighth at 148*26, and the 
ninth at 188*97 nearly. 
It is also worthy of remark that the second difference approaches 
closely to the value of |-7r 2 , namely to 4*93480, and we are tempted 
to conclude that this well-known number is the asymptote to which 
the second difference tends. The mere arithmetical coincidence is 
a weak argument in favour of this notion ; yet it is all that the 
algebraic formula seems capable of supplying: we shall find a much 
stronger argument in the character of the physical phenomenon 
under review. 
Lemma. 
The vibration of the portion LH, comprised between two cross- 
ings, is that of a musical string fixed at L and H, and stretched as 
by a weight HO at its lower end ; and the preceding investigation 
takes into account the change of strain due to the weight of the 
cord. In the case of the musical string the tension is many times 
greater than the weight of the cord, which weight, therefore, may 
be neglected even when the string is upright. 
Using as the linear unit the length of a pendulum oscillating in 
the same time as the string, and as the unit of tension the weight 
of one unit’s length of the string, and writing w for the tension 
so measured, the differential equation of the curve is 
— x = w . 2z x 
of which the solution in its most general form is 
. z z 
x =p . sin — — + q. COS——T 
SJW sjw 
where p and q are coefficients depending on the initial motion and 
on the elapsed time. In our present example q is zero, and the 
equation of the curve at some particular instant becomes 
Z 
x=p . sm — =-, 
V to 
which applies strictly to the case when the two ends are on one 
level. 
