242 



Mr. K. Lachlan. 



[Mar. 11, 



assisted in doing so by agitations effected either by thermal or 

 mechanical agency; hence — 



Rest after suspension, aided by oscillations at intervals, diminishes 

 the internal friction of a wire which has been recently suspended, or 

 which after a long suspension has been subjected to considerable 

 molecular agitation by either mechanical or thermal agency. 



On the contrary, when a maximum molecular rotatory elasticity 

 has been reached, molecular agitation, if carried beyond a certain 

 limit, diminishes the elasticity ; hence the results of " fatigue of elas- 

 ticity ;" and hence — 



Mechanical shocks and rapid fluctuations of temperature beyond 

 certain limits may considerably increase the internal friction, and, 

 though to a much less extent, diminish the torsional elasticity. 



The logarithmic decrement is independent of both the length and 

 diameter of the wire. 



II. " On Systems of Circles and Spheres." By R. Lachlan, 

 B.A., Fellow of Trinity College, Cambridge. Communicated 

 by Professor A. Cayley, F.R.S. Received February 23, 1886. 



(Abstract.) 



This memoir is an attempt to develop the ideas contained in two 

 papers to be found in the volume of Clifford's Mathematical Papers 

 (Macmillan, 1882), viz., " On Power Coordinates " (pp. 546 — 555), 

 and "On the Powers of Spheres " (pp. 332 — 336); the date of the 

 former is stated to be 1866, and of the latter 1868, but the editor 

 explains (see p. xxii, and note, p. 332) that though these papers 

 probably contain the substance of a paper read to the London Mathe- 

 matical Society, February 27, 1868, " On Circles and Spheres " (" Proc. 

 L. M. S.," vol. ii, p. 61), they were probably not written out before 

 1876. It is possible, therefore, that Clifford may be indebted to 

 Darboux for the conception of the "power of two circles," or spheres, 

 as an extension of Stein er's use of the "power of a point with respect 

 to a circle. Darboux was the first to give the definition of the power 

 of two circles, in a paper " Sur les Relations entre les Groupes de 

 Points, de Cercles, et de Spheres" ("Annales de l'ficole Nbrmale 

 Superieure," vol. i, p. 323, 1872), in which some theorems analogous 

 to the fundamental theorem of this memoir are proved. 



This memoir is divided into three Parts : Part I consists of the 

 discussion of systems of circles in one plane ; Part II of systems of 

 circles on the surface of a sphere ; and Part III of systems of spheres. 



The power of two circles is defined to be the square of the distance 

 between their centres less the sum of the squares of their radii. 



