1887.] E. Sang on Oscillations of Uniform Flexible Chain. 289 
derivative must come to exceed 0 , and it may be that the progression 
eventually becomes divergent ; into this matter we shall inquire 
hereafter. 
The first derivative ^ is expressed by the series 
1 Z £ 1 2 2 3 
~ I + l 2 2 ~ l 2 . 2 2 . 3 + 1 2 .2 2 . 3 2 . 4 “ ’ ^ c ‘ 
whose terms stand between those of the preceding in such a manner 
that, with scarcely augmented labour, one computation may be made 
£ 2 
to give both. Thus if we take a term of the series for x, say 
2 2 
and divide it by the next exponent 3, we get p 92 ^ a term of the 
z z 3 
series for lz x ; and this again multiplied by ^ gives - 2 -f 2 ^ 2 , the 
succeeding term of the series for x. 
To make a beginning, let us compute the values for z = 1 ; 
these are 
x= +. 22389, lz a;= -. 57672. 
On examining the relations of these numbers to unit and to each 
other by the method of continued fractions, we get a remarkable 
result for the ratio of the ordinate to its derivative. Proceeding in 
the usual way of taking the greater from the less, the remainder 
from the preceding, and so on, we get the quotients 
2, 1 ; 1? 2 ; 1,3; 1,4; 1,5; &c., 
whence the successive approximations alternately in defect and in 
excess. 
2 
1 
1 
2 
1 
3 
1 
0 
1 
2 
3 
5 
13 
18 
67 
85 
33 &c ' 
1 
0 
1 
1 
2 
5 
~7 
26 
and if we follow the method of excesses, the quotients come out 
1, 2, 3, 4, 5, &c., giving the chain of fractions 
1 2 3 4 5 6 
— 1_ 0 ! 2 5 18 85 492 
0" T T T 2 7 33 191 &C ‘ 5 
