PKOFESSOR C. J. JOLY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 2 G 7 
a quaternion array may be defined as a function which vanishes if, and only if, the 
constituents of every row can be linearly connected by the same set of scalar multi¬ 
pliers, It is multiplied by a scalar if every constituent in a column is multiplied by 
that scalar; and the sign of the array is changed if contiguous columns are trans¬ 
posed. 
These laws are precisely the laws which govern the function {abed), which is in 
fact a one-row array, so that if in (257) we replace any one of the four quaternions 
by any quaternion xa yh zo ivd, the quotient remains unchanged. The 
quotient is therefore an invariant in the Hamiltonian sense i it remains unchanged 
when the four quaternions a, b, c, d are operated on by the function Y. 
If we regard the lowest row as consisting of the results of operating by the 
special linear function unity on a, b, c and d, and if we replace by X/jY, 
X/ 2 Y, . . . X/,Y and unity in the last row by XY; to a factor, n^ 7 iy^, the quotient 
becomes the corresponding quotient for the system of functions 
xy;xr\ xy^x^-^... xy;x-h 
SECTION XL 
The Numerical Characteristics of Certain Curves and Assemblages 
OF Points. 
Pag, 
63. The number of points represented by {Qi, Q,2} = 0, Q;^ being a quaternion function of 
of order . 267 
64. The order and rank of the curve [Q1Q0Q3] ^0.267 
65. The order and rank of the curve ((QiQ2Q3Q4Q5)) = 0.269 
66. The number of points represented by (((QiQjQ3Q4QoQ,i))) = 0.270 
67. Conditions for the vanishing of the system [[Q1Q.2Q3Q4]] = 0.270 
03. In order to facilitate future investigations, we shall determine the numerical 
chaiactenstics of certain curves and systems of points which frecpiently occur. 
Using the symbol Q„ to denote a homogeneous ([uaternion function of y of the order 
M„, it appears from Salmon’s chapter on the “ Order of Restricted Systems of 
Equations ” in his ‘ Modern Higher Algebra,’ that 
{Q 1 Q 3 } — b, or -j- = 0 .(258) 
represents a system of points whose number is 
My + MyM^ + M^M^s + Mo3.(259). 
64. In like manner the chapter cited enables us to write down the order of the 
curve represented by 
[QiQaQs] = b, or -f = 0 .... (260) ; 
2 M 2 
