PEOFESSOR C. J. JOLY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 233 
• = ^P'lPz = Sp'ip'g = 0 ...... . (54). 
Hence in this order PiP^l^'iP'z is a quadrilateral situated on the unit sphere. 
These results may be verified for the vector form (44). 'Actually solving 
/{I + ot) = s (I + m) = e^— Sem + Yvt^, 
we see that s = — = Spe^, and therefore 
( 6 - —— S~'^Srje^, or (s^ — rj^)r^ = {s + r]) (e^ — S’^Spe^), 
so that operating by Se^ the result is tlie qnartic in s 
+ .s^(e/ — p2) _ (Spq)2 =0.(55) ; 
and fora real function two roots of this qnartic are always real and two are imaginary. 
Two of the united points are consequently real (Art. 7) and two are imaginary. 
SECTION II. 
The Classification of Linear Quaternion Functions. 
Art. 
13. Table of types and auxiliary formulte. 
14. Standard forms. 
15. Solution of the equation fq=p for functions of the first class. 
16. Case of functions of the second class. 
17. Functions of the third class. 
18. Functions of the fourth class. 
19. Self-conjugate functions. 
20. The classes of self-conjugate functions. 
21. If a function converts a tetrahedron into its reciprocal, it is self-conjugative . . 
22. Geometrical meaning of adding a scalar to a function. 
13. Linear quaternion functions may be classified according to the nature of the 
united points :— 
I. The first class consists of those functions which have no line or plane locus of 
united points, and it is divisible into sub-class :— 
I^, the four united points distinct. 
I 3 , two united points coincident. 
I 3 , three united points coincident. 
I4,, all four coincident. 
I5, two distinct pairs of coincident united points. 
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