PEOFESSOK C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 2^1 
The problem therefore reduces to the determination of g from the equation 
(compare (214)) 
// =/o¥/o“^-//-/u~W.(220). 
The form of this equation suggests the new function 
///=/o"t/’//u^ ///+/'// = 0. 
and the equation (220) reduces to 
./// = Cl' Ufa • 
( 321 ); 
(233); 
and the problem reduces to the determination of a function g commutative witlj tlie 
known function 
The function g must possess the same'i'' united points as ; or g must be of the 
form (compare (221)) 
g = x + + ?(’/;; ; g' .r _ ~ . , (223). 
Actually multiplying these expressions we find (219) 
pp' = 1 = [x + - (y/; + u;/;;i)i.(224); 
and as this equation must be equivalent to the latent quartic of the function 
it must vanish when for are substituted its latent roots. Now (Art. 23) the 
latent roots ofare identical with those of/), and the latent roots of the latter 
function (Art. 12) are of tlie form di dz \/ — Substituting and reducing, 
we find in terms of the two invariants nj' and of J], two equations 
1 = X' + (2y?r — 2^ nl'w~), 
0 = '2xz — y" + nj' {2giv — 2 ® + ngv'^'' 
. (225) 
connecting the four scalars x, y, z and iv. Hence, reverting to the original functions, 
the transformation 
F = .r + y/u 7;/o + -fo'Vi\fo + 77/o • 
. (226) 
converts the function / into itself; in other words, it converts the quadric and the 
linear complex 
S7//i = 0 .(227) 
into themselves. 
54. Passing on to the general case, let us consider the relations which must be 
satisfied when one function f can be converted into anotlier F ; or the conditions 
that a quadric and a complex can be simultaneously converted into another given 
quadric and another given complex. 
* Compare Art. 38, and the Appendix to Hamilton’s ‘Elements,’ vol. ii., p. 364. 
