Ball — Kinematics and Dyiiamics of a Rigid Syatem. 253 



This may be 



{U)x^ + {22) x^- + (33):r3^ + {A4.) x,^ + ((12) + {2\))x,x^, 

 or it is equally 



(ll)yr ^ (22)y,^ + (33)^3^ + {U)y^ + ((12) + {2\))y,y,. ' 

 The family of surfaces is accordingly 



u+\n = (). 



In the ordinary case of a rigid system in common space every cir- 

 cular cylinder whose axis is coincident with the screw about which 

 the system is displaced possessed this property. The system 

 U-\- AO = is thus the generalization to elliptic space of the system 

 of circular cylinders. 



The same result is obtained in another manner by calculating the 

 distance through which any point x is displaced when conveyed by 

 the transformation to y. 



Substitute Xi + Xy, &c., for Xi, &c., in the equation of the absolute, 

 and we find 



Xi + ^2' + ^3* + ^1' + 2X(xiyi + Xoy^ + x^y^ + x^y^) + X' (yi^ + y-^ + y^ + y^), 



or n + 2AZ7+ A.2f2 = 0, 



whence if ^ be the distance, we have 



cos 6 = -r. 

 O 



whence we see that the surfaces of the type 



. f^-n cos^ = 



possess the property that each of them is the locus of the point con- 

 veyed through the same distance $. In ordinary space we of course 

 notice that the points equidistant from the axis of the screw are con- 

 veyed through equal distances. 



We can now easily see the conditions that the general displacement 

 shall assume the special type of the vector. Cos must then be in- 

 dependent of X, a condition which implies 



(11) = (22) = (33) = (44) = cos^, 



and every equation of the type (12) + (21) = must be satisfied. We 

 then find that the displacement of every particle of the system is of 

 equal length. 



It will be easy by this analytical method to deduce all the general 

 properties of the motion of a rigid system in elliptic space. 



