151 
of Edinburgh, Session 1866-67. 
That is, the pressure at A must be proportional to the ordinate A a, 
the pressure at B to the ordinate B5, and so on. Now, if the 
ordinates A a, B6, Cc, D d, &c., were in arithmetical progression, 
there would be no flexure at any of the points ; the flexure is, in 
fact, proportional to the second difference of the ordinates ; for 
example, the flexure at D is proportional to the second difference 
of the three ordinates Cc, D d : Ee, or to Cc — 2D d + Ee. Again, 
when a straight bar is slightly bent by pressures applied at three 
points, the flexures at equidistant intermediate points are in arith- 
metical progression ; that is to say, if at the point D no pressure be 
applied, the flexure at D must be an arithmetical mean between 
those at C and E. When pressure is applied at D, that pressure 
must be proportional -to the second difference of the flexures at C 
and E, and consequently to the fourth difference of the ordinates 
at B, C, D, E, and E. 
Hence the characteristic property of the form assumed by such 
a discrete series when making a simple ’vibration is, that the length 
of any ordinate, as dJ), must be proportional to the fourth difference 
of the five ordinates of which it is the middle one. 
In passing from the discrete series of masses to the concrete 
bar, the differences of the various orders are replaced by the deri- 
vatives of the same orders, and hence the general character of the 
curve assumed by an elastic bar when making a simple vibration 
is this, “ that the ordinate at any point of the curve is proportional 
to its fourth derivative, the abscissa being the primary;” and this 
statement contains the complete solution of the problem. 
Putting x for the abscissa, and y for the ordinate, of the curve, 
the above proposition may be written 
txV = ; 
it indicates the relation between the function and one of its de- 
