266 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
sought to show that the expression for the product of their squared differ- 
ences was inherently positive. This he succeeded in doing by transforming 
the said expression into a sum of squares, the result being reached by 
proceeding from particular to general, and by a combined process of guess 
and test. The equation being 
or, say, 
a - x h g 
h b - x f 
= 0 , 
g f c-x 
x z - Vx 2 + Q# - R = 0 , 
the expression referred to is* 
P 2 Q 2 - 4P 3 R + 18PQR - 4Q 3 - 27R 2 ; 
and Rummer’s equivalent for it is 
15 2j[(fh(b-c)+JX9*-h>)Y 
+ 2 [ 2 ( 6 ~ c )( c - «)A + (2c - o - b)fg + (2 h 2 -/ 2 - # 2 )7(] 2 
+ [(6 - c)(c - a)(a -b) + (b- c)f 2 + (c - a)g 2 + (a - b)h 2 ] 2 , 
o 
where 2 indicates the summing of the expressions obtained by performing 
simultaneously the cyclical substitutions 
fa b c \ 
( f 
9 
h\ 
m b c a J > 
h 
Jacobi, C. G. J. (1844, March). 
[Sulla condizione di ugualianza di due radici dell’ equazione cubica, 
dalla quale dipendono gli assi principali di una superficie del 
second’ ordine. Giornale Arcadico, xcix. pp. 3-11: or Crelle’s 
Journ., xxx. pp. 46-50 : or Gesammelte Werke, i. pp. 271-276.] 
By using A, B,. . . for bc—f 2 ,ca — g 2 , . . . Jacobi first puts Kummer’s 
sum of squares in a neater form, namely, 
152,(.<7 h - ^G) 2 + 2,( 6F -/ B + cF - 2aF + 2 A) 2 
+ (6C - cB + cA - aC + aB - bA) 2 , 
and he then gives a lengthy but thorough verification of its accuracy. 
From the fundamental identity 
ax 2 + by 2 + cz 2 + 2 fyz 4- 2 gzx + 2 hxy 
= L(a^ + + y Y z) 2 + M(a 2 a + p 2 y + y 2 z) 2 + N(a 3 a + fay + y & z) 2 , 
* I.e. - ^ of what afterwards came to be called the discriminant of x 3 - Px 2 y + Qxy 2 - Ry 3 . 
