1902-3.] Dr Muir on Axisymmetric Determinants. 559 
These all indicate most important advances, and, be it noted, 
the last of them is given by its author as the third of a series the 
law of which he considered “ facile a saisir.” In view of this, and 
the fact that in the following year he published his great memoir 
containing the multiplication-theorem in all its generality, we are 
bound to conclude that whatever credit in the latter he must share 
with Cauchy, the axisymmetric case of it is entirely his own. 
Further details need not be given, as this has already been done 
in Part I. of this history. 
Jacobi (1827). 
[Ueber die Hauptaxen der Flachen der zweiten Ordnung. Cretins 
Journ.-, ii. pp. 227-233. 
De singulari quadam duplicis integralis transformatione. Crelle’s 
Journ., ii. pp. 234-242.] 
In these two papers, which owe their inspiration to the famous 
memoir* of Gauss on the “ Determinatio Attractions . . . 
Jacobi concerns himself with two problems of transformation, the 
first of which explicitly deals with the transformation of the 
ternary quadric 
Ax 2 + B y 2 + C z 2 + 2 ayz + 2bzx + 2 exy 
into the form 
L£2 + + N£ 2 , 
and the other implicitly with the corresponding change in the case 
of a quaternary quadric. The papers will be fully discussed when 
we come to deal with “ determinants of an orthogonal substitution.” 
It suffices for the present to note that in the first Jacobi virtually 
gives as an equivalent for the axisymmetric determinant which we 
should now write in the form 
x - A x cos v — c x cos fx - b 
x cos v - c x - B x cos A. — a 
x cos y. - b x cos A. - a x - C 
* Commentationes societatis regice scientiarum Gottingensis recentiores, iv. 
(1818) : or Gauss, JVerJce, iii. pp. 331-355. For abstract see Gottingische 
gelehrte Anzeigen (1818, Feb.), pp. 233-237 : or WerTce , iii. pp. 357-360. 
