288 Proceedings of Royal Society of Edinburgh, [june *jo, 
Each succeeding term of this progression is formed from its 
z z z 
antecedent by means of a factor of the form 7r„ , ^ 5 ••••—», so 
J 22 32 n l 
that, however large 2 may be taken, the denominator n 2 must 
eventually come to exceed it ; and thus, although the terms may 
increase at the beginning, they must ultimately come to decrease ; 
and therefore the computation of x to within any prescribed degree 
of precision is always possible. 
The curve lies on the one or on the other side of the plumb-line, 
according as the sum of the even terms of the progression exceeds 
or falls short of the sum of the odd terms, and we can discover 
which way only by the actual calculation. The intersections of 
the curve with the middle line represent the points of suspension 
of the oscillating chain, and therefore our attention is first called 
to the discovery of those values of 2 which correspond to a? = 0. 
From the mere aspect of the progression we could not even predict 
that any such values are possible, or form any idea of the order of 
the roots of this transcendental equation. The consideration of the 
physical problem with which it is connected does indeed throw 
light on the matter, and leads us to anticipate an endless succession 
of roots more and more separated as we proceed upwards. 
The accurate determination of these roots can only be reached by 
trial; the computations are very operose, and we look for some 
means for lessening the labour ; this is found in the law of succes- 
sion of the derivatives. Let us suppose that, corresponding to some 
value of 2, the ordinate x and its derivative lz x have been computed, 
we are then able easily, particularly if 2 be represented by an integer 
number, to deduce the subsequent derivatives. Thus — 
2*®= 
w 
r-H 
1 
l 
2 
3 z^ = 
— lz x — 2 . 2 z x 
Z 
~~ 2 — ^ * 3 ^ 
iz x — 
* 7 . '■ 
and so on ; 
and these enable us to deduce the values of x corresponding to 
proximate values of 2 by the process described in the work above 
referred to. When 2 is large these values evidently decrease at the 
beginning, but as the order advances the multiplier of the last found 
