NOTICES AND ABSTRACTS 



OF 



MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS. 



MATHEMATICS AND PHYSICS. 



Mathematics. 



The Parallelogram of 3Iechanical Magnitudes. 

 By HoMERSHAM Cox, B.A. 

 The following six kinds of magnitude considered in mechanical science, — 

 Of translation — Of rotation — 



Statical forces. Statical couples. 



Linear velocities. Angular velocities. 



Linear accelerations. Angular accelerations, 



all conform to the remarkable and well-known law, that if two sides of a parallelo- 

 gram represent the magnitude and direction of two components of either kind, their 

 resultant is similarly represented by the diagonal. 



It is here proposed to show that this law may be proved for the six kinds of mag- 

 nitude indifferently by a method of demonstration common to them all. It is clear 

 that such a demonstration can be derived only from fundamental principles common 

 to them all, and it is verj^ interesting to trace out these axioms, which may be re- 

 garded as the abstract causes of the coincidence of the laws of the effect of mechani- 

 cal agents of nature so diverse as those above specified. Extraneous and unneces- 

 sary considerations, imported into the particular demonstrations, are excluded from 

 the general demonstration, which involves no empirical axioms, and deduces its re- 

 sults solely from analytical and geometrical principles, and the definitions of the 

 measures of the magnitudes considered. 



The direction of a couple, angular velocity, or angular acceleration is here defined 

 to be the direction of the axis about which either respectively tends to turn, or turns 

 a system of material parts, of which the relative positions do not vary. The mea- 

 sure of the relative magnitudes of couples is the relative magnitudes of their mo- 

 ments. The measure of the relative magnitudes of angular velocities is the relative 

 magnitudes of the angles through which they respectively turn in a given time, a 

 system of material parts of which the relative positions do not vary. The measure 

 of the relative magnitudes of angular accelerations is the relative magnitudes of the 

 angular velocities which they respectively generate in a given time. The other terms 

 used in this demonstration do not here require definition. 



From the definitions of a Resultant, and of the measures of the six kinds of mag- 

 nitude above specified, it may be shown that — 



I. The resultant of components having the same direction is their algebraical sum. 

 Therefore (II.) two equal and opposite components destroy each other's effect. 



The inclination of the resultant of two components inclined to each other at a 

 given angle is independent of the unit by which they are measured, and depends 

 solely on their relative (not absolute) magnitude — conversely. 



(III.) The inclination of the resultant of two components inclined at a given angle 

 determines their relative magnitude. 



1851. 1 



