260 PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEO.METRY. 
SECTION IX. 
Tue Determixatiox of Linear Transformations which satisfy Certain 
Conditions. 
Arfc. 
52. Transformations converting one given quadric into another. 
53. The transformations of a quadric and a linear complex into themselves. 
51. The conditions that it may be possible to transform simultaneously a given cpiadric 
and a linear complex into another given quadric and a linear complex. 
55. Transformations converting one conic into another given conic. 
5G. The condition for the simultaneous transformation of two conics into others 
57. One twisted cubic may be converted into another with arbitrary correspondence of the 
points. 
58. The condition for the conversion of one unicursal quartic into another. 
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52. The results of Art. 47 afford a simple solution of such problems as to find a 
transformation ivlncli shall convert one quadric into another. 
Symbolically this problem amounts to solving the equation 
Fi=.rF„/.(214) 
which connects two known self-conjugate functions Fj and Fo with an unknown 
function / and its conjugate. 
I he first quadric is reduced to the unit sphere by the transformation 
l/i- = so that SqFir 2 = Sqp.(215). 
The unit sphere is converted into itself by the transformation (Art. 47) 
= f/'in so that ^qf —Sqf it qq' = 1 .(216); 
and finally the sphere is converted into the second quadric by the transformation 
so that Sqy = SqgFpyg.(‘-^17). 
Thus the transformation 
/= FA“f/Fi% qq' = 1 .(218) 
converts the first quadric into the second; and evidently this is the most general 
transformation fulfilling the conditions. 
53. To convert an arhitrary function “/” ^nto itself observe that the transforma¬ 
tion must belong to the group (compare (218)), 
F =/o ; where / = /o +/, qf I .(216), 
v hich converts the self-conjugate part /g of the function into itself. 
