PROFESSOE C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 241 
22 . It may be as well to show the geometrical meaning of changing from a 
function f to another f — as in Art. 14. 
Writing 
P' = if- ^o) Q = V - hP P =fl .; 
it is obvious that p' is some point on the line pq. To determine the point, let 
p', P and Q be the points p>', p and q with unit weights, then 
p/ _ V — tf ,q _ fq — _ pS/ q — t ,,q 
Sp — t^^Sq S/q — Iq S/q — 
and we have the ratio of segments 
p^ _ Q — p' _ S/q 
P P P — P' tQ 
(109); 
( 110 ), 
or its ratio is directly proportional to the perpendicular from the point Q on the 
plane S/q = 0, which is projected to infinity by the transformation.''^ 
Hence it is eas}" to form a geometrical conception of the nature of a transformation 
by reducing it to some simpler type, as in Art. 14 ; the point P for instance may 
always be supposed to lie in a fixed plane, while in the case of functions of the 
classes II and III it may lie supposed to lie on a fixed line or to l^e a fixed point. 
SECTION III. 
Scalar Invariants. 
Art. 
23. The extent of the invariance. 
24. The sum of the latent roots is zero. 
25. The sum of fractional powers of the roots is zero. 
26. Tetrahedron inscribed to one cpiadric and circirmscribed to another 
27. The sum of the prodircts of the rood's zero. 
28. The sum of the roots and the sum of their reciprocals zero . . . 
29. Twelve-term invariants. 
23. From the results of Arts. 5 and 6, it appears that 
((/- t)a, (/- t) h, (/- t) c, if- t) d) = {ahcd) (n - n't + ~ n'"P + P) ( 111 ) 
is identically true, no matter what the value of t may be or Avhat quaternions 
C6, b, c, and d. may represent. In this sense the four scalars n, n', n", and n'" are 
invariants, and every relation connecting them implies some peculiarity in the 
geometrical transformation produced f. 
* In vectors, if q = 1 + p, the ratio is d + Se'p). = to~^xTe if x is the length of the perpendicular. 
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