238 UNIVERSITY OF VIRGINIA PUBLICATIONS 



If A be changed to d and this equation integrated 



s= 2fc,- Si, 



which is BernouiUi's theorem connecting the arc lengths of curves traced 

 by points Pi moving in such a manner that at each instant the tangents at 

 the points are parallel. 



The whole theory of homogeneous strains admits of very beautiful and 

 symmetrical treatment through this displacement theorem, we follow it no 

 further here, however, than to point out the enunciation of a simple theorem 

 in a homogeneous plane movement. 



Three points A, B, C determine the homogeneous movement of the 

 points of a plane in a fixed plane by revolving in circles with equal angular 

 velocities in the same direction about three centers Ci, d, Cg in the fixed 

 plane. The radii of revolution being n, r2, rs then any point P whose areal 

 coordinates with respect to ABC are x, y, z describes a circle with the same 

 angular velocity about the point x, y, z with respect to CidQa and its radius 

 is given by 



r- = 2a;ri- — 'Zxyrn^. (18) 



The segment ra are constants and are the segment differences of n, r2, rz, 

 the angles between the latter are constants. The relative figure is a rigid 

 figure rotating about its fixed origin. The locus of points in the original 

 plane which describe circles of equal area is an ellipse and by varying the 

 area a family of similar and similarly situated ellipses. The centers of these 

 circles in the fixed plane lie correspondingly on a similar family of ellipses 

 in the fixed plane, the one family being a homogeneous displacement of the 

 other. They have a common center which is at rest, its x, y, z coordinates 

 are the same as those of the fixed origin in the relative figure. The circles 

 ^ described by all points on the same straight line envelope a conic, the radii 

 of which points at any instant envelope a parabola. 



If one of the points, say C, revolves with the same velocity but in the 

 opposite direction the area (C) is negative. Any point (not C or on the 

 straight line AB all of which describe circles) in general whose coordinates 

 are x, y, z with respect to ABC describes an ellipse whose center with respect 

 to C1C2C3 is z, y, X and whose radius vector sweeps over equal areas in 

 equal times. Points describing paths of equal area lie on an ellipse, hyper- 

 bola or parabola according as the radius n of (C) is less than, greater than 

 or equal to the radius of the minimum circle described by the points of the 

 line AB. In either case by varying the area there results a system of simi- 

 lar and similarly situated conies, all points on one of which describe paths 



