286 Proceedings of Roy cd Society of Edinburgh. [june 20 , 
g 5 x dPx 
and — , whereas the mark x v can show none. The writer, in 
dz 5 dip 
his “ Solution of Equations of all Orders,” Edinburgh, 1829, has 
placed Lagrange’s accents as ante-subponents, and has written along 
with them the primary, so that the symbol S2 x is used to denote the 
fifth derivative of x regarded as a function of z , while 5 jc means the 
corresponding differential coefficient when y is the independent 
variable. In this way all the essentials are exhibited without 
redundancy. 
The symbols x , lz x, 2 z x, 3 z x., thus indicate a series of functions 
deduced by the repeated operation called differentiation ; each one is, 
as Lagrange says, the derivative of the preceding, and each one is 
the primitive (integral) of the succeeding. So we may carry the 
notation backwards by using the sign of reversion and write _ lz x for 
the function of which x is the derivative, — that is the fxdz of' 
Leibnitz, and thus get the progression extending both ways 
-32'^-' ? -2 2 «, -12^ > } 1 2*U 22*^ 5 32*b &C. 
Using this notation, the condition of a simple oscillation of the 
chainTs expressed by 
- ax = l2 a? -f 2 . 2 z x . 
From this equation we have the second derivative in terms of x , 
of the first derivative lz x, and of the primary z ; and from it also we 
easily obtain the subsequent derivatives, for on differentiating we find 
— a . lz x = 2 . 2z x 
+ 
z . 
32 ^ 5 
CL • = ~ 3 • QzpC 
+ 
z . 
42 ^ 5 
55 
II 
55 
CO 
1 
+ 
z . 
5 z X > 
and in general - a . nz x = (?i+ + z . (n+2)z x ; we may also pro- 
ceed backwards by integration, thus : — 
and in general 
a . 
1 
II 
0 
X 
+ 
z . 
a. 
- 22^ — 1 • 
-la® 
+ 
z . 
X 
a . 
- 32 ^= - 2 . . 
-22^ 
+ 
z . 
-1 Z X 
— a . 
- nz^ 
= - in- 1 ) 
• -(«- 
-DzPO + Z. 
55 
N 
cT 
1 
3 
1 
cf>x 
to 
represent some 
one 
of 
these : 
equation relating to it will take the general form 
- a. <f>z = n. lz 4>z +- 2 : . 2 Z cf>z 
