1887.] E. Sang on Oscillations of Uniform Flexible Chain. 295 
Thus it seems that the portions AB, DE, GH, KL are convex 
toward the axis. 
The values of z corresponding to these points of reflexure are the 
roots of the transcendental equation 2z x = 0, and are obtained in the 
manner already described ; they, along with the corresponding 
values of x, are 
z 
X 
6-59365 
- -13228 
17-71250 
+ -06448 
33-75518 
- -04001 
54-73005 
+ -02792 
80-63878 
- -02090 
while the distances of the points of reflexure below the respective 
crossings are 
1*02416 
1-00925 
1-00489 
1-00303 
1-00206 
The details connected with these singular points, namely, the 
crossings, the maxima, and the reflexures are contained in the sub- 
joined table : — 
Singular Points 
in 
the Curve. 
z 
X 
lzX 
2 gX 
0-00000 
00000 
+ 
1-00000 
ooooo 
1-00000 
ooooo 
+ -50000 
ooooo 
1-44579 
64903 
•ooooo 
ooooo 
— 
•43175 
48070 
+ •29862 
83407 
3-67049 
26605 
— 
•40275 
93957 
•ooooo 
ooooo 
+ -10972 
89745 
6-59365 
41007 
- 
•13227 
94874 
+ 
•13227 
94874 
•ooooo 
ooooo 
7-61781 
55861 
•ooooo 
ooooo 
+ 
•12328 
26057 
- -01618 
34589 
12-30461 
40804 
+ 
•30011 
57525 
•ooooo 
ooooo 
- -02439 
05051 
17-71249 
97297 
+ 
•06448 
25277 
— 
•06448 
25277 
•ooooo 
ooooo 
18-72175 
16977 
•ooooo 
ooooo 
— 
•06273 
64998 
+ -00335 
09952 
25-87486 
34727 
- 
•24700 
48771 
•ooooo 
ooooo 
+ -00965 
04810 
3375517 
72165 
— 
•04000 
79701 
+ 
•04000 
79701 
•ooooo 
ooooo 
34-76007 
11118 
•ooooo 
ooooo 
+ 
•03942 
82580 
- -00113 
42974 
44-38019 
17035 
+ 
•21835 
94072 
•ooooo 
ooooo 
- -00492 
01997 
54-73004 
72864 
4 - 
•02791 
85486 
— 
•02791 
85486 
•ooooo 
ooooo 
55-73307 
59044 
•ooooo 
ooooo 
— 
•02767 
63754 
+ -00049 
65880 
67-82041 
35683 
— 
•19646 
53715 
•ooooo 
ooooo 
+ -00289 
68471 
80-63877 
90738 
— 
•02090 
51560 
+ 
•02090 
51561 
•ooooo 
ooooo 
81-64083 
82279 
•ooooo 
ooooo 
+ 
•02077 
67294 
- -00025 
44894 
