OF THE EOOTS OF SYSTEMS OF EQUATEOHS 
525 
73. This is right, and shows (clianging sign) that the well-known expression 
(discriminant) 
he —f ^ — h(f — ch^ + 2 fyli, 
is equal to 
— (20 dl^) + (ir Id dl) — (d2 Id^), 
a form which, for the ternary quadric, vanishes at sight. 
Another form is 
'^20^3 T" 
74. The expression 
4a20<^03 d~ ^11^10%! 
satisfies the partial differential equation which appertains to the differences of the 
quantities in the relation 
{1 + a^x-\- /3yj) (1 -f = 1 + -f + « 022 /^- 
This equation is 
2 + 2 9«„ + o^oi 9^^, = 0 . 
It is not, as a fact, expressible as a function of differences of a^, ^i, a. 2 , ySg, because it 
vanishes altogether for a fundamental identity of the order 2. 
In relation to a fundamental identity of order greater than 2, the expression does 
not satisfy the equation of differences. Although it may be regarded from the above 
as a vanishing function of the differences, it is convertible into a non-vanishing 
function by the transformation before given. The transformed expression is 
(2d d2) - 2 (Iff) or (S 30 S 02 ~ ^ 11 ^). 
which visibly satisfies the differential equation 
9x0 + f'oi = 3^10 + 
§ 14. The Construction of Symmetrical Tables. 
75. From the first law of symmetry it has been established that it is possible to 
form two symmetrical tables in connexion with every partition of every biweight. As 
illustrations, I give certain of the results as far as weight 4, inclusive. We have 
presented for the weight 4 the biweights 40, 31, 22, 13, 04. The theory of the 
biweights 40 and 04 is jirecisely the same as that of the weight 4 in the uni partite 
theory.* The one is, in fact, concerned only with the single system of cjuantities 
* Vid.e ‘American Journal of Alatliematics,’ vol. 11, and succeeding volumes. 
