A CERTAIN QUADRATIC FORM 237 



the invariable plane of a conservative system. Holditch's theorem is there- 

 fore the lowest degenerate form of Laplace's beautiful generalization. 



5. Let 



Q=ix' -x'r+iy' -y'T- + ■ ■ ■ 



We are concerned here with a homogeneous displacement (strain) of 

 the points of space. 



If the points P,- be displaced in any manner through segments ASi 

 to a new position of configuration P,-, the k's remaining constant, the point 

 P is displaced through a segment As to a new position P'. The ^'s are 

 the coordinates of P with respect to P; and also of P' with respect to Pi'. 

 The segment As is the vector sum of the segments fciAs; and its value 

 is given by 



As2 = Sfci Asi^ - I,kikj Asii^, (17) 



where Asj,- is the relative displacement of the points Pj and P,-. We observe 

 also that As,-,- is equal to the vector difference of PiPj and P/P/, as well as 

 that of As.- and As,. 



The relative figure is constructed by drawing from a fixed origin segments 

 parallel and equal to Asi, the segments joining the ends of these two and 

 two are the segments As,-,- forming a figure with respect to which Asi are 

 the radial coordinates of the origin and fc; are the coordinates of P where 

 OP = As. With reference to the relativfe figure (17) shows that the locus 

 of points corresponding to points of equal displacement is a circular locus, 

 and a concentric family of such when As varies. In the original space the 

 locus of points of equal displacement is shown by (17) to be a closed quad- 

 ric surface and the variation of As gives a family of similar and similarly 

 situated such surfaces. There is a point in the original space which has a 

 zero displacement, it is the center of the family of closed quadrics, audits 

 coordinates with respect to P,- or P/ are the /c-coordinates of the fixed 

 origin in the relative figure. In the relative figure if a straight line be 

 drawn through the origin then all points on this straight line have their 

 displacements in the same direction and are equal to their distances from 

 the zero point. Hence given the displacement of three points and the zero 

 point the direction and magnitude of the displacement of an arbitrary point 

 is at once constructed by an easy straight line construction which is quite 

 obvious. If the displacements ASi are parallel the right side of (17) is a 

 perfect square and 



As = 2fci As.-. 



