[659 
660] 
153 
»riginal group is 
>on the 3 letters 
= 2, and thence 
a- i b 1 c 1 a. 2 b. 2 c. i : viz. 
)./ = also 
are a^boC^a^Cx, 
OA) (6iC 8 ) (c^), 
)e the order of 
! a submultiple 
is even, quasi- 
of two quasi- 
e product of a 
it follows hence 
tlf of them are 
i the g terms 
rder, or it may 
regarded it is 
stop to prove), 
ed of ^ cycles, 
re according as 
;al substitution 
odd; and the 
g as ~ (a — 1), 
tv 
, the term A, 
.ve substitution 
e manner with 
660. 
ON THE COBBESPONDENCE OF HOMOGBAPHIES AND 
POTATIONS. 
[From the Mathematische Annalen, t. xv. (1879), pp. 238—240.] 
It is a fundamental notion in Prof. Klein’s theory of the “ Icosahedron ” that 
homographies correspond to rotations (of a solid body about a fixed point) : in 
such wise that, considering the homographies which correspond to two given rotations, 
the homography compounded of these corresponds to the rotation compounded of the 
two rotations. 
Say the two homographies are A + Bp 4- Gq + Dpq = 0, A 2 + B 2 q + G x r + D 2 qr = 0, 
then, eliminating q, the compound homography is A 2 + B 2 p + C 2 r + D 2 p?'= 0, where 
A 2 , B. 2 , G 2 , D^BxA-AxG, BxB-AxD, DxA-GxG, B 2 B - GxD ; 
and the theorem is that, corresponding to these, we have rotations depending on the 
parameters (X, g, v), (X 1} g 1} zq), (\ 2 , /¿ 2 , v 2 ) respectively, such that the third rotation 
is that compounded of the first and second rotations. The question arises to find the 
expression for the parameters of the homography in terms of the parameters of the 
corresponding rotation. 
The rotation (X, /¿, v) is taken to denote a rotation through an angle ^ about 
an axis the inclinations of which to the axes of coordinates are f, g, h, the values 
of X, g, v then being = tan ^ cos/, tan cos g, tan ^ cos h respectively: (\ 1} fix, Vx) 
and (X„, /i 2 , Vo) have of course the like significations; and then, if (X, g, v) refer to 
the first rotation, and (X 1} fix, Vx) to the second rotation, the values of (X 2 , g 2 , v 2 ) 
for the rotation compounded of these are taken to be*: 
X 2 = X -f Xj + gvx — fixv, 
g. 2 = g + fix + vXx — VxX, 
v 2 = v + Vx + Xgx — Xxg, 
* The numerators might equally well have been X + X x - (vv 2 - Vx v )i e tc., but there is no essential difference r 
we pass from one set of formulae to the other by reversing the signs of all the symbols: and hence, Im 
properly fixing the sense of the rotations, the signs may be made to be + as in the text. Assuming this 
to be so, if we then reverse the order of the component rotations, we have for the new compound rotation 
the like formulas with the signs - instead of + ; but this in passing. The formulae, virtually due to 
Rodrigues, are given in my paper “On the motion of rotation of a solid body,” Camb. Math. Journal, 
t. m. (1843), [6]. 
C. X. 
20