Lateral Vibration of Bars supported at Two Points. 497 
in none of these cases is the p.d. Hg | electrolyte necessarily 
zero. 
4. It is shown that the Paschen relation immediately fails 
when the chemical action at the drop-electrode due to 
atmospheric oxygen becomes appreciable. 
5. The Paschen E.M.F. between mercury and a solution 
of KC1 remains unchanged on the addition to the solution of 
small quantities of Na 2 S, although the natural p.d. between 
mercury and the KC1 solution is thereby altered by more 
than half a volt. 
6. A critical percentage of Na 2 S was found for which the 
natural p.d. just mentioned altered with extreme rapidity 
as in the cases studied by Behrend, in which he observed th@ 
variation of the E.M.F., Hg | Hg 2 (N0 3 ) 2 | Hg, produced 
by the gradual addition of KC1 or KBr at one electrode. 
In the present case the change in the p.d. Hg ] electrolyte 
was deduced directly from the electrocapillary curves. 
XL VI. On the Lateral Vibration of Bars supported at Two 
Points with One End overhanging. By John Morrow, 
M.Sc., D.Eng.; Lecturer in Engineering, Armstrong College 
( University of Durham)*. 
WHEN a bar is carried on two supports and has one end 
overhanging by an amount c, as shown in the figure, 
a a 
I 
^! 
its natural period of vibration is to be determined from the 
equation 
(cosh ml sin ml — sinh ml cos ml) (cosh mc sin me — sinh mc cos me) 
— 2 sinh ml sin ml (1 + cosh me cos mc) = 0, 
in which 
m = (27tN)*(/dg>/EI)*; 
E = Young's Modulus for the material ; 
I = the moment of inertia of the section about an axis 
perpendicular to the plane of bending ; 
a = sectional area of bar (assumed uniform) ; 
p = density of material ; 
N = frequency of natural vibrations. 
* Communicated by the Physical Society : read November 8, 1907. 
