962] 
COORDINATES VERSUS QUATERNIONS. 
543 
And for the comparison of the two solutions, we have 
a = i (x 2 - x x ) + j (y 2 - 2/1), /3 = i (x 4 - x s ) +j (y 4 - y 2 ). 
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 
we 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 + kz, where i, j, k are the 
Hamiltonian symbols (i 2 —j 2 = k 2 = — 1, jk = — kj = i, ki = — ik=j, i — —ji = k)\ 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. 
Coordinates. 
Taking 0 as origin, the coordinates of 
A, B, C are taken to be 
Oi, Vi, *i), O2, y 2 , z 2 ), (x, y, z) 
respectively. Then 
xx x + yy 1 + zz x = 0, 
xx 2 + yy 2 + zZo = 0 ; 
whence 
x : y : z = y x z 2 — y 2 z x : z,x 2 - z 2 x, : x x y 2 - x 2 y x . 
Quaternions. 
Points A, B, C are determined by their 
vectors a, ¡3, y. Then 
Say = 0, $/3y = 0 ; 
whence 
my = Fa/3, 
m being an arbitrary scalar. 
Here to compare the two solutions, observe that the two equations Say = 0, S/3y = 0 
are in fact the equations xx x + yy x + zz x = 0, xx 2 + yy 2 + zz 2 = 0 ; and so also my = Fa/3 
denotes the relations x : y : z — y x z 2 — y 2 z x : z x x 2 — z 2 x x : x x y 2 — x 2 y x . But a quaternionist 
says that my = Va/3 is the compendious and elegant solution of the problem as 
opposed to the artificial and clumsy one x : y : z — y x z 2 — y 2 z x : z x x 2 — z. 2 x x : x x y 2 — x. 2 y x . 
And it is upon this that I join issue; my= Fa/3 is a very pretty formula, like the 
folded-up pocket-map, but, to be intelligible, I consider that it requires to be developed 
into the other form. Of course, the example is as simple a one as could have been 
selected; and, in the case of a more complicated example, the mere abbreviation of 
the quaternion formula would be very much greater, but just for this reason there 
is the more occasion for the developed coordinate formula. To take another example,