52 EE. B. Wilson, 
P; (a, d) = k (d—a? 
and cannot vanish unless a and d are equal. In the third case the 
determinant is 
(OO) Naa y 3a?x + 2ay—z 3ex+ay—azt+w «| 
ax+2 38e@y+2az—w 3a y+a?z—aw BA ey 
azt+w 3a2+2aw 302+ a2w ol eee 
aw 3aw 3a w w 
where the 4 m, p have been omitted as they obviously go out by 
the same reasoning as before. Here the w factors out of the last 
row, which may then be multiplied by «2, vy, ¢ and subtracted re- 
spectively from the first, second, third rows. The w then factors 
out of the third row, which may be multiplied by vy, z and subtracted 
from the rows above. Now the factor w drops out of the second 
row, which may be multiplied by zg and subtracted from the top row 
whereupon the y drops out. The condition is reduced to 
(100’) Onru Oa rly eee 
0 a ——— 
an eee Gt AOm ew, aN 
| Gy sa isa ok 
which is clearly unfulfilled. 
The last case is instructive, because it illustrates the dependence 
and independence of different repeated roots. The condition is 
(OD | Garey “ai(@ 20) —62y C2061. 6) Cee | 
ay Qa 20) a) a(2ac4-c*)y Bae 
ci-pw ele Zaye aw c(2ac.-@)e--¢ 2a 
cw c(¢e+2a)w c (2ac + a’) w w 
The y, w of the second and third lines go out, and a reduction 
similar to that given before removes the x, z. It is this possibility 
to get rid of the coefficients in the expression for 6, which shows 
that these coefficients may have any values other than 0. The 
condition reduces to 
(101) 1 15): 2 0 
BAGG) nd (2aC Cat : 
1 J a 0 | a 
[kG welsh 2m) Ve@ee taza | 
which amounts to merely 
(101°) P, (a, ¢) = k (a—c)* 
and cannot vanish. 
The foregoing cases are typical of all that can arise. With regard 
to the case where z has any value the following remarks will 
suffice. Suppose that 2 has & repeated roots’ a. Construct the 
half square upon the portion of the main diagonal of 2 correspond- 
