1887.] E. Sang on Oscillations of Uniform Flexible Chain , 303 
in which the denominators are the continued products of the terms 
of two arithmetical progressions whose common difference is s. 
The cases of n being zero, or being a negative integer, demand 
special consideration. When n is zero the above general formula 
would give cf>z = A (1 — oo ) ; but this infinity does not exist in the 
nature of the problem — it has been introduced by our mode of pro- 
cedure. The original condition - acfjz = z. 2z (fiz may belong to a 
cc 
variety of physical problems, as to this one 2t x = a-^, in which the 
incitement to motion is proportional to the distance directly and to 
the time inversely. In this case we get 
<f>Z = B | 
z 
- a A = 1 . 0 . B , 
- aB = 2 . 1 . C 
- aC = 3 . 2 . D 
- aD = 4 . 3 . E 
a 22 a 2 £ 3 
A = 0, B = B , 
- a 
C 
D = 
E = 
1.2 
- a 
273 
- a 
3~4 
w 
B , 
C , 
D , &c. 
z 4 
1 1 1.2 T 1.2 1.2.3 1.2.3 1.2.3. 4 + &c * } 
When n = 1 we have 
- a A = 1 , ( - 1 ) . B 
- aB = 2 . 0 . C 
- aC = 3 . 1 .D 
-aD = 4. 2 . E 
A = 0 
B = 0 , C = C 
- a 
r) = L3 C ’ 
E = 
- a 
2.4 D, &c. 
, nrt f z 2 a 2 3 a 2 Z i ft3 ( s n ) 
1 1.2.3 + 1.2 1.2. 3.4 ~ 17273 1.2. 3.4.5 H eSC '/ 
And similarly for n= - 2 
T O , T, f ZS « 24 , g2 t < l 
'"• 3 1.2.3 1 1...4 + 1.2 1....5 &C ’ } 
where we at once observe that each series is the primitive (integral) 
of the preceding. 
VOL. xiv. 14 /11/87 u 
