1887.] E. Sang on Oscillations of Uniform Flexible Chain. 305 
and thence the equations of condition, 
a A = %B 
aB = (2w+1.2/>)C 
aC = (3 n + 2 . 3p + 1 . 2 . 3)D 
aD = (4 n + 3 . ip 4- 2 . 3 . 4)E 
aE = (5^ + 4.5p + 3.4.5)F , &c., 
which may be written 
aA = n. IB 
aB = (n+ \p)2Q> 
aC = (n + 2p + 1 . 2)3D 
eiD = (n + 3p + 2. 3)4D , 
and in general 
aQ = + (r - 1 )p + (r - 2)(r - l)}rP , 
and thus the denominator in the term containing a r z r is the con- 
tinued product of the natural numbers 1.2.3. up to r, multiplied 
by the continued product of all the terms of the progression 
(n-p + 2) + r(p — 3) + r 2 
for all values of r up to the same limit. 
The terms of this progression can be resolved into products when- 
ever n and p are such that (p - \) 2 - in is a square number, and 
then the denominators are the continued products of the terms of 
three arithmetical progressions. 
Thus when p — 3 and n = 1 , that is when the condition ax = lz x + 
3z . ^ + z 2 . 3z x is proposed, the solution becomes 
x — A 
- az a 2 z 2 
1 + 1 
l 3 1 3 .2 3 
+ 
a 3 2 3 
1 3 .2 3 .3 3 
+ &c 
■ } 
in which the three progressions coincide. 
On introducing fractional coefficients we get arithmetical pro- 
gressions with differences other than unit ; thus, on making a — ^, 
2 
n = g p — 2 the three progressions become 
1, 4, 7, 10, &c. ; 2, 5, 8, 11, &c., and 3, 6, 9, 12, &c., 
whose terms just fill up the progression of natural numbers, so that 
for the condition 
