56 
Proceedings of the Royal Society of Edinburgh. [Sess. 
The investigation, due in essence to Lagrange (1759),* is based on a 
simple transformation, three examples of which we may write for ourselves 
in the form 
a Y 
= (ayc + a^ 2 + 
(aye + a 2 y + a#) 2 + 
I 1 2/ 2 ’ 
y z 
l«AI i«AI 
I <h C \ I I «iC 3 I 
y 
z 
^1 d> 2 <^3 
a? 
y 
z 
w 
(ayc + . • • + a 4 t 0 ) 2 
?/ g w 
I ^1^2 1 I ^1^3 1 I ®iA I y 
a i c 2 Wh C s\ \ a l C ±\ 2 
I a i^2 I I I I a A I W 
where, it must be borne in mind, the square arrays are axisymmetric.f By 
dividing both sides by a x it is seen that in the case of the binary quadric 
the conditions are 
aj>0, | a-fi 2 1 >0 . 
In the next case it is equally evident that they are a x >0 and the like 
conditions for the quadric in y,z: and as by the previous case the latter 
are 
|«AI > 0 , 
«A I I a A I 
> 0 
I a \ C 2 I I a i C Z I 
we obtain in all for the ternary quadric 
a 1 > 0, |<*AI >0, \a l b 2 c z \>0. 
Similarly for the quaternary quadric we must have a v \a 1 b 2 \, \aA c s\’ 
| aA c z d± | all positive : and so on, the last determinant in each case being the 
discriminant of the quadric. It is casually added that if the 1st, 3rd, 5th, 
. ... of the series be negative and the others positive, the quadric will be 
negative for all real values of the variables. 
As might be expected from the connection with Lagrange, it is also 
pointed out that, calling these determinants A v A 2 , A 3 , . . . , we can by 
repeated applications of the fundamental transformation change the n-ary 
quadric into the form 
AiU n 2 + + 
+ 
^2-U, 2 
1 
* See immediately preceding footnote. 
t This restriction may be done away with if we alter the squared expressions on the 
right hand into 
(a 1 cc + a 2 ^)(a 1 cc + & 1 ?/) , (a x x + a 2 y + a 3 z)(a l x + b 1 y + ^z) , .... 
