TRANSACTIONS OK THE SECTIONS. 49 



swinging bar describe compound vibration-curves of the form known as Lissajous's, 

 of great regularity and distinctness, and was suggested to the author by Mr. 

 Newall as a new means of tracing them. In the new instrument the horizontal 

 bar is hung by four strings forming a W ; and the outer pairs are nipped together 

 at equal distances from the rod at whatever height above it gives the desired 

 period of its longitudinal vibrations. Its transversal vibrations are of two kinds, 

 either of bifilar torsion, or of simple lateral oscillation about the three upper points 

 of suspension. The points of attachment on the bar are a little above its axis, which 

 passes through the centre of gravity of a large fixed weight at its middle point ; 

 two smaller sliding weights, moved along it, regulate the rate of its angular oscilla- 

 tions. The new pendulum possesses a fourth mode of vibration — of rotation round 

 the line of attachment in the bar, like the rolling of a ship at sea — a condition of 

 oscillation very similar to one which was lately ingeniously employed to illustrate 

 that problem by Sir William Thomson. If the bar " rights " quickly round this 

 axis, these small rolling oscillations do not accumulate very greatly, and soon dis- 

 appear ; but if they are nearly of the same period as the principal transverse vibra- 

 tion, thej' are so large and persistent as entirely to disturb the regularity of the 

 curves. A glass pen fixed to the end of the bar traces Lissajous's cm-ves by com- 

 bining the longitudinal with either of the two transversal vibrations. When both 

 of the latter act together, wavy modifications of Lissajous's cun'es are produced, 

 which present cusps, stationary points, and other interesting varieties of form [of 

 which some illustrations were exhibited]. Their general expression is given oy 

 the equations 



\ x = A cos (a -\- at) 



(«/ = B sm{b+^t) + Csm(c+yt), 



which only differ from those of Lissajous's curves by the addition of a second in- 

 dependent term at the end of the last equation. 



On the Influence of Temperature on the Elastic Force of certain forms of Springs. 



By F. H. Wenham, 



The author stated that the value of springs in the form of elastic plates or rods 

 subject to deflection or torsion, in the construction of instruments for measuring 

 and regulating force, temperature, and time, depends upon the law that the degrees 

 of motion are equal to the forces, and that this equality of force and motion is 

 identified with the tiine in which those motions are performed ; for the vibration of 

 certain forms of springs is performed in the same time, whether the degi-ee of motion 

 is great or small : such a spring will give the same musical note at all ranges, and 

 have the important property of isochronism, as illustrated in the balance-springs of 

 chronometers, meaning that the time is the same at all ranges in the arc of \'ibration. 

 The author pointed out that the form of balance-spring commonly used in time- 

 pieces is not strictly isochronal ; for bej'ond one revolution the forces are unequal, 

 increasing during winding and decreasing in the opposite extreme of uncoiling, but 

 that in the acting range of vibration of these instruments the differences were not 

 appreciable. 



Instruments for measuring force, temperature, or time, such as aneroid barometers, 

 thermometers, or chronometers, the accuracy of whose indications depends upon the 

 uniform elasticity of springs, require a compensation to counteract the loss of elas- 

 ticity by increase of temperature. A number of experiments were tried and detailed 

 by the author, in order to determine a law to enable the compensations to be 

 effected definitely. The materials experimented upon were steel, hardened and 

 tempered, crown-glass, brass, and germau silver highly condensed by hammering. 

 These materials, while under various degrees of compression, were subjected to 

 temperatures ranging up to 500° ; but it was found that the loss of elasticity did not 

 correspond in a regular ratio with the increase of heat; for example, in a steel 

 spring each hundred degrees from 100° to 500° caused deflections in the ratio of 13, 

 16, 40, and 52 ; and, in first experiments, when the springs had cooled they did not 

 return to their normal point with the pressure remaining the same, but had acquired 



1873. 4 



