STUDIES IN STATISTICAL REPRESENTATION. 493 
three indices being taken instead of two. Subject tosome 
limitation this may be made to pass through six points 
besides the origin. | 
For the determination of the constants it will be necssary 
to take the six points so that the values of the abscissae 
will be in geometrical ratio, as with the four points in the 
preceding case. This gives then six equations which, 
analogously to the previous case, may be written :— 
y= L+M+N 
y, = La + MB + Ny 
yz; = Lo? + MB? + Ny? 
ys = La? + MB? + Ny’ 
By reasoning similar to that in the preceding case, it 
may then be established that «, 6, y, are the roots of the 
equation 
J 1 Yo Ys | = On..eereees (50) 
GY, Sar Ya 
3 4-4 YS 
Ga ve. Us) Ye 
which may be expanded in the form 
ae SA +d Ase = TA, = OC antes es (50a) 
where A,, — 3A,, 3A,;and — A,are the minors respectively of 
x’, x x, and1inthe determinant. The roots of this equa- 
tion must be real and positive. 
The condition that the roots should be positive is that. 
A,, Az, A;, A, should have the same sign. 
To examine the equation for real roots it must first be 
deprived of its second term. It then becomes 
X*+3 X (A,A, — A’) +3 A,(A,A, — A) + AB - AjA,=0......... (51) 
The equation y’ + qy + r = 0 will have its roots real and 
unequal if ey + a) isnegative. The condition becomes 
in this case 
3 A,(A,A, — AZ) + A} - AA, 
9 
ad 
2 
t + (4,4, —- A$)? is negative. 
