1893-94.] Prof. Cayley on Coordinates mrsus Quaternions. 273 
themselves equal and parallel, viz. (writing to denote equal and 
parallel), if AB ry. CD, then AC o. BD. 
Coordinates. 
Quaternions. 
A, B, C, D are determined by 
their coordinates 
Hi)’ i^2’ (^ 3 ’ y^’ 
AB no CD gives 
Xn X-t Xa ^ , 
^ ^ ^ ^1, whence 
y^-yi^y^-Vz’ ^ 
^3 - ^1 = ^4 - ^2 1 
y^-y\=y^-y2’ 
that is 
AB, CD are determined by 
their vectors a, j3, and then 
writing y for the vector AD. 
AB ^ CD gives a = /?, 
whence 
y_/5= - a + y, 
that is 
AC ~ BD. 
AC a, BD. 
And for the comparison of the two solutions we have 
a = «(«2 - *l) +J{$/2 - 2']). /3 = *(^4 - %) +i(y4 - %) • 
But this example of a plane theorem is a trivial one, given only for 
the sake of completeness. 
Passing to solid geometry, we have — 
Coordinates . — Considering a fixed point 0, and through it the 
rectangular axes Ox, Oy, Oz, the position of a point is determined 
by its coordinates x, y, z. But w^e may, in place of these, consider 
the quadriplanar coordinates (x, y, z, w) linear functions of the 
original rectangular coordinates x, y, z. 
Quaternions . — The position of a point in reference to the fixed 
origin 0 is determined by its vector a, which is in fact = ix -\-jy + Jcz 
(where i,j^t^ are the Hamiltonian symbols (^2=y2 = ^2^ _ 2^ 
jli = -hj- i, M = - ih =j, i — -ji ■■= Itj ) , but the idea is to use as 
little as possible the foregoing equation a = ix +jy + kz, and thus to 
conduct the investigations independently, as far as may be, of the 
particular positions of the axes Ox, Oy, Oz. 
I consider the problem to determine the line OC at right angles 
to the plane of the lines OA, OB. 
VOL. XX. 14 / 7 / 94 . 
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