PEOFESSOR C. J. JOLT ON QUATERNIONS AND PROJECTIVE GEOMETRY. 263 
-f 2 Soyiz) + (^' 1^11 + %^ 12 ) ^1 
-f- ~|~ 2 -1“ ^’ 0 ^ 22 ) ^2 "1“ (^1^21 '^ 2 ^ 22 ) ^'2 ^ • ( 233 ) 
'-''' 2 ^ 12 ) “ 1 “ ^ {^\y\i “ 1 " ^^ 2 ^/ 12 ) “ 1 “ (*' 11 “t“ 2 ^ 12 )^ 1 
+ “ 1 ” ^'-2 ^'22)^^'2 H“ 2 ('^’^?/^21 “t“ '^'21/'22) ^^'2 “t“ “ 1 “ *‘^^^22) ^ 2 ~ ^ ’ 
ill which 5 i, So, s\, s'o are arbitrary, but the other scalars given may be taken as 
determining the two pairs of relations connecting the two sets of six quaternions. 
When the left-hand members of the equations analogous to (232) are multiiilied by 
-f s^a-jg, &c., and added, the sum is zero; and the sum of the right-hand members 
is (with an obvious abbreviation) 
{ (^‘l^<^ll + ^•2a^J2, ^’1^11 + «2?/l2^ ^l^l 1 + + ^I^i^’lIwV'l)^'l + I?«'lV'] fc'l } 
“b {(^i^' 2 ] “b ^ 2 ^ 22 ’ '^ 1^21 ~b ^ 2 ?/ 22 ’ '^ 1%1 ~b '^ 2 ~ 22 X^ 2 ^ 2 )'^^ 2 2 ^ 2 ) ^2 ~b 
Xwb'w'o T^' 2 ] — ^.(234), 
or, for simplicity, 
(s^Xji -f- -S2X12) + 2 (s^Y]^]^ + S2YJ3) l> I + (si^Z^j + s 2212) c ] 
“b (s^Xoj + S 2 X 02 ) “b 2 (S]^Y2]^ + SgYoo) 6b. -b (^i + 5220 . 2 ) cb= 0 . . (235) 
where X^^ is a quadratic in iq and Y^^ Z^^ its successive polars to u\v\. This rela¬ 
tion connecting the six quaternions must be equivalent to the second equation (233), 
so we may equate corresponding coefficients of quaternions, when we shall obtain six 
equations linear in Sj, Sg, s\, s' Let 6‘b and 5 3 be eliminated from them. The result 
is the system of determinants 
^ii'®i“bX]o6‘2 YyS2"bYio53 7j^yS-^-\-Zii^s^ Xo^^^d-Xo.o.So Yo^s^-bYg.oSg Zo^Si+Zggsq 
y i\ ^11 ^21 y 21 
y 12 ^12 ^22 y 22 
X 
11 
X 
12 
21 
2 ' 
^ 22 
= 0.(236), 
which is equivalent to four equations. But 6 q and S 3 are arbitrary; consequently this 
system of determinants breaks u]i into two independent systems, equivalent to eight 
equations among the eight scalars u, v. The eight scalars enter homogeneously into 
the equations, and may be eliminated, leaving a single coiidition connecting the four 
conics, in order that it may be possible to find a transformation which shall convert 
two of them into the remaining two. 
57. A twisted cubic may he transformed into another twisted cubic ivith arbitrary 
correspondence of the points. 
The equation of transformation of one arbitrary twisted cubic into another is 
(compare Art. 43) 
f{abcd'ft, 1 )^ = {a'b'c'd'Jut -f- u, vt + v')^ .(237). 
Hence equating coefficients of t, four equations are obtained which serve to deter- 
