1887 .] E. Sang on Oscillations of Uniform Flexible Chain. 285 
The resolution of our physical problem is now converted into the 
management of a case in the doctrine of functions, and thus it 
acquires an importance far beyond any that the original question 
can be supposed to possess. In this equation z stands as the 
dx dix 
primary variable, x as its function, as the first, as the second 
derivative. Translated into geometry, we have the abscissa z, the 
doc d 2 x 
ordinate x , the inclination of the curve , and the curvature j 
di 
dx 
or into mechanics, the time t (for z), the position x, the velocity jj > 
d 2 oc 
and the acceleration all combined in one formula; and the 
resolution of it may imply that of whole classes of physical pro- 
blems. It is in this light that the matter is again brought under 
the notice of the Society. 
The problem does not belong to the differential calculus, for in 
that case we should need to have the relation of the primary z to 
one of the derivatives explicitly declared; not to the integral 
calculus, for then the connection between the primary and the 
derivative would need to be given ; nor yet to the third co-ordinate 
branch, for the relation of the primitive and derivative functions is 
not prescribed. 
For convenience in treating the matter, it is expedient to discard 
Leibnitz’ notation for differentials of higher orders than the first, 
and altogether to dispense with his notation of integrals. 
dix 
Such an expression as -j-g, is intended to represent the result of 
five successive differentiations in which z is the primary or inde- 
pendent variable and x the function. Here the sign of differentiation 
is twice, and that of the order also twice, written. How, the 
essentials to be indicated are, the idea of differentiation, the 
primary, the function, and the order. The idea may conveniently 
be indicated by the position of the marks ; Lagrange placed these 
as accents over the function, thus : — x\ x 11 , x ul , x lv , x v . This scheme 
has two drawbacks; the position of the accents had long been 
appropriated to the indices of powers, and there is no notice of the 
primary ; Leibnitz’ notation clearly shows the distinction between 
