64 



Proceedings of the Royal Irish Academy. 



II. — Displacement and Eotation. 



9. The displacement of a rigid schema in n dimensions is evidently ex- 

 pressed by au orthogonal transformation with constants added : — 



x = a + liX + ni^y + UiZ + piU + . . . . 



y' = 1 + l^x + 7)Uy + «2» + Pi'U' + • . ■ ■ 

 s' = C + hx + Vhy + TlsZ + paU + . . . . 



u' = d+ liX + viiy + TiiZ + piW + . . . . 



etc., , 



where a, h, c, d, etc., arc constants. Also 



'2P = 1 and Slrh = horizontally, and (^the equivalent) 

 ^P = 1 and Ilrm,. = 0, etc., vertically. 



Evidently distance is preserved in this transformation. Note also that it 

 involves — — ^ — independent constants. 



10. Condition of continuity.— It A denote the determinant 



li nil ih T9\ 

 Iz nil iH ^2 



li nii TOi ^4 



then A' = 1. 



The condition that the transformation he continuous and therefore capable of 

 heing precisely represented hy a displacement of a rigid schema is 



A = + 1. 



For if the transformation is identical (x = a;,' etc.), A = + 1, and A cannot 

 suddenly jump from + 1 to - 1. Discontinuous transformation is exemplified 

 by reflexion with regard to an (S',,.,, e.g. if n = 3, 



x' = - X, y' = y, z' = z 



is discontinuous, as it would turn a right-hand glove into a left-hand. 



11. Rotation and rotatory flats. — A rotation round an S,„ is defined as a 

 rigid-schema movement in which the points of Sm are kept fixed, while the 

 other points in Sn move with one degree of freedom. The possibility of a 

 rotation round a; = 0, y = is shown by the transformation 



x =x cos Q + y sin d, y' = - x sin + y cos ^, 



the other co-ordinates being unaltered. 



