154 
ON THE CORRESPONDENCE OF HOMOGRAPHIES AND ROTATIONS. 
[660 
each divided by 
I XXj jJLfX-y , 
and if we then write for X, p, v, the quotients x, y, z each divided by w, and in 
like manner for X x , v 1 and \ 2 , p 2 , v 2 , the quotients x 1} y x , z 1 each divided by w x , 
and x 2 , y 2 , z 2 each divided by w 2 , the formulae for the composition of the rotations are 
X 2 — XW x + X x W + yz x — y x z, 
y2 = yWi + ViW + zx x - z x x, 
z 2 = zw x + z x w 4- xy x — x x y, 
w 2 = ww x — xx x — yy x — zz x ; 
and the question is to express A, B, G, D as functions of (x, y, z, w), such that 
A x , B x , C x , D x denoting the like functions of (x X) y x , z x , w x ), A 2 , B 2 , C 2> D 2 shall 
denote the like functions of (x 2 , y 2 , z 2 , w 2 ). 
It is found that the required conditions are satisfied by assuming 
A, B, G, D = ix — y, —iz + w, —iz — w, —ix — y, 
(where i — V— 1 as usual) : in fact, we then have 
A 2 =B x A-A x G 
= (- iz x + w x ) (ix -y)- (ix x - y x ) (- iz - w) 
= i (xw x + x x w +yz x - y x z) — (yw x + y x w + ZX x - z x x) 
= - 2/ 2 + ix 2 , 
as it should be: and we verify in like manner the values of B 2 , G 2 and D 2 respectively. 
The result consequently is that we have the homography 
(ix —y) + (— iz + w) p + (— iz — w) q + (— ix — y)pq = 0 
, x y z 
where - , —, — 
z 
- are the parameters of 
corresponding to the rotation ( -, -, - J : 
^ ° \w ’ w ’ w) ' 
rotation, tan cos /, tan ^ cos g, tan ^ cos h. 
I remark as regards the geometrical theory that, if we consider two lines J and K 
fixed in space, and a third line L fixed in the solid body and moveable with it; 
then, for any given position of the solid body, the three lines J, K, L are directrices 
of a hyperboloid, the generatrices whereof meet each of the three lines: and these 
generatrices determine, say on the fixed lines J and K, two series of points corresponding 
homographically to each other: that is, corresponding to any given position of the 
solid body we have a homography. But it is not immediately obvious how we can 
thence obtain the foregoing analytical formulae. 
Cambridge, 3 April, 1879.