470 Proceedings of Royal Society of Edinburgh. [july 18 , 
to his memoir Recherches d' Aritlimetique (1773: see above), viz. 
“les nombres qui peuvent etre representees par la formule 
B£ 2 + Ctu + D«/ 2 .” 
Gauss writes bis form of the second degree thus — 
axx + 2b xy + cyy ; 
and for shortness speaks of it as the form (a, b, c). The function 
of the coefficients a, b , c, which was found by Lagrange to be 
of notable importance in the discussion of the form, Gauss calls the 
“ determinant of the form,” the exact words of his definition being 
“ Numerum bb - ac , a cuius indole proprietates fornne 
(i a , b, c ) imprimis pendere in sequentibus docebimus, deter - 
minantem huius formse uocabimus.” (xv. 2.) 
Here then we have the first use of the term which with an extended 
signification has in our day come to be so familiar. It must be 
carefully noted that the more general functions, to which the name 
came afterwards to be given, also repeatedly occur in the course of 
Gauss’ work, e.g. the function a§ - /3y in his statement of Lagrange’s 
theorem (xxn.) 
b'b' - ac = (bb - ac)(a 8 - /fy) 2 
But such functions are not spoken of as belonging to the same 
category as bb - ac. In fact the new term introduced by Gauss 
was not “ determinant ” but “ determinant of a form,” being thus 
perfectly identical in meaning and usage with the modern term 
“ discriminant.” 
Notwithstanding the title of the chapter Gauss did not confine 
himself to forms of two variables. A digression is made for the 
purpose of considering the ternary quadratic form (“ formam 
ternariam secundi gradus”), 
axx + axx' + a"x"x" + 2 bx'x" + 2b' xx" + 2b" xx, 
or as he shortly denotes it 
( a, a\ a"\ 
b , b\ b'j . 
In the matter of nomenclature the following paragraph of this 
digression is interesting 
“ Ponendo bb — a' a" — A, b'b' - aa" = A', b"b" — cia' = A”, 
ab-b'b" = B, a'b'-bb"= B', a"b" -W = B", 
