( 230 ) 
which both the other squares are equal to zero. If the coefficient 
of y? is negative, then a locus exists (a conic section) which indicates 
: - : Urn : : J 
all mixtures for which 2=—” has the same value. If this locus is 
xy 
reduced to one point, as is the ease if the coefficient of y* vanishes, 
then 2 is for that point a minimum, respectively a maximum. The 
-minimum value of 2 satisfies therefore the equation: 
5 25/, - ; ; 
(a, —4b, (a, —4b,)—(4,,—4D, ,) $ (a, 4b, )(a, —b,)—_(a,, —4b,,)"|— 
4 | (a Es 
— f(a, —Ab,)(a,,—Ab,,)—(4,,—4,.)(4,,; —49, 5); = 9, 
or | a, —Ab, , a,,—A),,, 
4) a, —Ab, , a,,—Ab,, ing OAT en 
dis —Âbiss %3—A4b,,, a, —Ab, 
a,,—Ab,, | 
a,,—4b,, ’ 
For the determination of rz and y we have moreover the equation : 
(a, —ab,)(1—a—y) + (a,,—Ab,,)u + (a,,—6,,)y — 9 
and the equation, which follows from the other square when it is 
equated to zero. 
Another way in which we might have reduced the equation 
zy—Abx, = 0 to the sum of three squares, would have yielded the 
following two equations for the determination of rv and y. 
(a,,—Ab,,)(1—#—y)+(a, —Àb, )e+(a,,—2b,,)y = 9 
and (a,,—Ab,,)(1—«—y)+(a,,—Ab,,)¢+(a, —Ab, )y = 0. 
Eliminating 1—vr—y, « and y from these three equations in which 
they. occur linearly, we find again equation (2). 
In order to calculate « and y we may derive the following rela- 
tions from these three equations. 
lay = y 
41, —4b,, : a,,;—Ab,,|  |a,;—20,; , a,-—Ab, | |e —Àb, , a,,—4b,,| 
| bee, é | 
a, —ab, , obs zg Raabe s alb es a » a, = 2b, |: 
or 
1—w«-y D y 
a, —Ab, , a,,—Ab,, © [Gan Abas » a,,—Ab,, eae VEE 
a,,—A4b,,, a, —Ab, | la, —2b, , a,,—ab,,| la, abe » @,,—Ab,, 
and 
1—a—y es a y 
Ass dbs ‚ a, —Aab, | [4s —ab, , aa Abal lage edn —4b,| 
> | 
ladi 5 as = bi; | B se ae ee 
In order that 2 have a minimum value for positive values of w, y 
and 1—w—y the following relations must hold: 
eg! 
ere Tet 
