436 HISTORY OF THE THEORY OF DETERMINANTS 
algebraical problem differed from that dealt with in the earlier 
paper of the same year merely in having four independent 
variables instead of three. Using modern phraseology, we may 
say that the one paper dealt explicitly with the transformation 
of a ternary quadric into the form L^ 2 + + N£ 2 , and the 
other implicitly with the transformation of a quaternary quadric 
into the form G^ 0 2 + G'^ 2 + G"^ 2 2 + G"'f 3 2 ; and such being the 
case, it is a matter for some surprise that the consideration 
of the corresponding problem for an n-ary quadric was left to 
Cauchy. 
In the lengthy paper we have now come to, the algebraical 
problem is no longer kept in the background, but forms one 
of the three parts into which the subject-matter naturally 
divides itself. The first is the “ Introductio,” occupying §§ 1-9, 
pp. 253-264, and containing a brief account of previous related 
work, followed by an indication of the new results reached. 
The second is headed “Problema I.” and occupies §§10-15, 
pp. 264-279, its subject being an algebraical transformation pure 
and simple. The third and longest is headed “ Problema II.” 
and concerns the closely related, not to say dependent, problem 
of the transformation of a double integral. With this clear-cut 
subdivision there is no need for any process of sifting: we turn 
at once to Problema I. 
It is stated by Jacobi as follows:—Proponitur, per substi 
tutiones lineares 
x = «s + a s' + a" s" w = at + bu + cv 
y = (3s + /3V + /3"s" w' = a't -f b'u + c'v 
z = ys + y's + y"s" w" = a"t + b"u + c"v 
quae identice efficiant 
x 2 + y 2 + z 2 = s 2 + s' 2 4- s" 2 , 
w 2 + w' 2 + w" 2 = t 2 + u 2 + v 2 , 
transformare expressionem 
(Ax + By + Cz)w + (Ax + By-f C'z)w' + (A^'x + B'y + C^jw" 
in hanc simpliciorem 
Gst + GVu + G"s"v.