621 
AND IMAGrINAKY BOOTS OF EQUATIONS, 
become determinable by means of the same cubic equations, viz. 
03_ K 02 + KWL0_ L2=o . 
g, cr, t will be to each other as the roots of this equation ( 34 ). 
(41) Since ru 5 -\-sv^-\-tu 5 represents a function in x, y with real coefficients, it follows 
that when L is positive, u, v as well as w being real, a : (3 : y are ratios of real quantities, 
and the roots of the preceding cubic will be real ; when L is negative, u, v becoming 
conjugate imaginary functions of x, y, whilst w remains real, r, s, unless they are equal, 
must become conjugate imaginary constants. When r, s, t are all real, g, a, r will be so 
too ; and when r, s are imaginary and t real, g, a will be imaginary and r real. Thus 
according as L is positive or negative the roots of 0 are or are not all real. Hence 
understanding by A the discriminant of the preceding equation with respect to 6 and 1, 
■jy must be always either zero or negative. We see a priori that must be integer, 
because when L=0 the cubic has two equal roots, To compute its value more con- 
veniently, write K=6#, J=12 \j. Then the equation becomes 
(1, 2*, W-jL, L 2 X0, -l) 3 , 
of which the discriminant is 
L 4 + 4(3£ 2 -;L) 3 + S2k 3 L 2 -12k 2 (3k 2 -jL) 2 -12icL%U 2 -jL). 
Hence 
£=L 3 — 108^'+ 36^/L-4/L 2 +32^L 
+ 72tj-12k 2 fL -36£ 3 L+12/£L 2 
=L 3 — 36£*/+24F/L- 4/L 2 - 4/; 3 L+12j/TL 2 . 
Accordingly, multiplying the above equation by — 3T2 2 in order to avoid fractions, 
replacing Jc,j by their values in terms of K, J, and naming G the quantity —432 
( 34 ) For since the absolute values of g, a-, r are not in question, we may consider §, <r, r as the roots of 
— K0 2 +29— r, so that § + <r+r=K. We have then 
jW 
which gives r= L 2 . Again, 
( ? <rr) 3 ( ? + er + r) 3 K 3 
(K 2 ~4 qf 
r 
As regards the sign to be given to JL in q, since 
ftrrK 2 _ K 2 
(K 2 — 4 2 ) 2= P’ or 
r L 2 
° r K 3 ~K 3 ’ 
J 2 , or (K 2 -4 2 ) 2 = L 2 J 2 , or . 
we have (K 2 — 42') 3 =J 3 L 3 . Hence 
Consequently 
J 3 (K 2 — 4§) 3 _(K 2 — 4q) 3 
L~ r 2 D ’ 
k 2 -Hjl 
K 2 -JL , K 2 + JL 
q =. — ^ — , and not 
4 o 2 
4 
4 
