1901-2.] Dr Muir on the Theory of Orthoyonants. 247 
connection with determinants, viz., two by Jacobi in 1827, one by 
Cauchy in 1829, three by Jacobi in 1831-3, and one by Catalan 
in 1839. 
Jacobi (1827). 
[Ueber die Hauptaxen der Flachen der zweiten Ordnung. 
Crelle’s Journ ., ii. pp. 227-233.] 
Without unnecessary preliminaries Jacobi enunciates the problem 
which he wishes to solve, viz., the transformation of an expression 
of the form 
Ax 2 + By 2 + Cz 2 + 2 ayz + 2 bzx + 2 cxy , 
where a?, y , z are the coordinates of a point referred to an oblique 
coordinate-system, into an expression of the form 
L £ 2 + M ^ 2 + ]S T £ 2 , 
where £, y, £ are the coordinates of the same point referred to a 
rectangular system having the same origin. This implies that the 
things directly sought are the nine coefficients which give each of 
the original coordinates in terms of the new. 
Jacobi, however, prefers to begin with a related set of unknowns, 
taking the equations which give the new coordinates in terms of 
the old. These being assumed to be 
£ — ax + Py +yz, 
7] = a'x + py + yZ , 
£ = a "x + p'y + y"z, 
the equivalent set giving the old in terms of the new is of course 
c 
1 (py'-p'y)i + (P'y-Py")y + (Py'- : fi'y)£) 
A-y = (ya - y"a)£ + (y"a - ya)y + (ya - ya)£ > 
A‘Z ■■= (cl ft" - CL "p)£ + (a"/3 - a p')rj + (a p - a P)£ j 
where 
A = OLpy" + (3y a" + ya p' - a p'y - p/'a - ya "p. 
Denoting the known angles between the original axes by A, y, y, 
there is obtained at once the set of six equations 
