146 Prof. Baker, On transformations with an absolute quadric 



Conversely, it is an easy matter to prove directly that the two 

 inversions (H-i) {H), about points on AB, are together equivalent 

 to a rotation about the line DC, of amplitude equal to twice the 

 Cayley separation of H and H-^^, in regard to the quadric, with a 

 similar fact for (K-^) {K). This gives an elementary theory of the 

 transformation. 



In Euclidian space, a general movement of a rigid body is thus 

 obtainable by a succession of four inversions, two of these about 

 points on the central axis, and two on the line at infinity of the 

 plane at right angles to this. Taking the central axes to be a; = 0, 

 y = 0, the two former may be taken to be reflexions in planes 

 z = k, z = kj^, and the two latter to be reflexions in the planes 

 X = 0, X = y tan ^6. By the former we obtain 



z -\- Zj_= 2k, z' + z-j_= 2^1, 

 giving z' = z + 2 {kj^ — k); hj the latter we obtain 



Xi= - X, ^1 = y, {x'~ Xj) sin ^d + {y'- y^ cos Id = 0, 

 {x'+ x-^ cos \d — {y'+ yj) sin ^d = 0, 

 giving 



x'+ a?! cos 6 — yx sin ^ = 0, y'— y-^ cos 6 — x-^ sin ^ = 0. 

 Thus altogether 

 x'— X cos d + y sin 9, y'= — xsind + y cos 6, z'= 2 + 2 (^j— k). 



