304 Proceedings of Royal Society of Edinburgh, [june 20, 
In our original inquiry the tension at any point was measured by 
the length of the chain below it, but if we suppose a load w to be 
appended, that tension will be represented by w + 2 , and the condi- 
tion necessary for a simple oscillation will be expressed by 
-ax = lz {(w + z) lz x } = lz x + (w + z) , 
and then the investigation becomes much more complex. On using 
the same method as before we find 
- ah = l 2 .B + 1.2w.C; C= _ aA + 12 - B ; 
1 .ziv 
- aB = 2 2 .C 4- 2.3w. D; D = ; 
Z.ow 
4- V- D 
-«C = 3 2 .D + 3 Aw. E ; E= - 0 + - i 
6 Aw 
in which each new coefficient is deduced from the two preceding 
ones. In this case the computation of the coefficients is very tedious, 
while that of the function and of its derivative is more so ; but 
these being found for any given 2 , the successive derivatives are 
found as before, for 
— a.x — lz x 
— (w + z) 2z x , 
ct • 12,0c ~~ 2 0 2 zpc 
= (w + z) 3x x, 
& • 2 ^ • 3 
= (w + z)^x , 
and thus it is easy to compute the value for any proximate value 
of 2 . 
The complexity in the preceding case arises from the circumstance 
that the terms of the series for w. 2z x are displaced from those of 
z. 2x ; and that, therefore, each new coefficient involves the two pre- 
ceding ones. Now the terms of lz x } z.x, z 2 . 2z x, &c., are all coinci- 
dent, and thus we are tempted to go one step farther and to propose 
for inquiry the condition 
ax — n. lz x +PZ.&.X + z 2 . 2 ; x . 
Proceeding in the same way as before, we have 
x = A 
+ Bz 
+ 
Cz 2 
+ 
D Z 3 + 
Ez 4 + 
la? = B 
+ 2Cz 
+ 
3Dz 2 
+ 
4Ez 3 + 
5Fz 4 + 
2 . <i z X = 
1 . 2Cz 
+ 
2,3Dz 2 
+ 
3.4Ez 3 + 
4.5Fz 4 + 
2 ™ _ 
. z Jj 
1. 
2. 3D z 2 
+ 2, 
.3.4Ez 3 + 3.4.5Fz 4 + 
