534 On the so-called Tschirnhausen Transformation [49 
Since 2/1 + 2/2+ ••• +2/« = 0, 
- 2B 2 = - 22y 2 y 2 = 2/, 
so that it is the character of 2y 2 which determines the projective character 
of B 2 . The number of real values of y is the same as of x. Hence if / has i 
pairs of imaginary roots, 2y 2 will be the sum of n — i positive and i negative 
squares of real linear functions of u lf u 2 , ... u n _ 1# 
Consequently, by virtue of the lemma above proved, there is only one 
element of uncertainty as to the character of 2y 2 , that is, it must we know 
a priori, when reduced to a sum of n — 1 positive and negative squares of 
linear functions of u 1 , u 2 , ... u n ._ 1? contain either i or i— 1 negative squares. 
This uncertainty may be removed by means of a second lemma, namely, that 
the discriminant of B. 2 is a numerical multiplier of the discriminant of/. 
When two of the roots of /are equal, two of the values of y become equal 
so that 2y 2 becomes reducible to a sum of n — 2 instead of a sum of n — 1 
squares. 
Hence the former contains the latter as a factor: moreover it is obvious 
from the form of each value of y that its discriminant regarded as a function 
of the n roots of /will be of the degree 2 {1 + 2 + ... + (n — 1)}, that is, 
n(n — 1) which is the same as that of the squared product of the differences 
of the roots of/. Hence B 2 is & numerical multiplier of such squared product. 
To find the value of the multiplier, I observe that in general it follows from 
known algebraical principles that if F is a sum of the squares of n linear 
functions of n — 1 variables the discriminant of F may be found as follows. 
Form an oblong matrix with the coefficients of the several linear functions. 
The determinant represented by what Cauchy would have called the square 
of this matrix, but which is more correctly to be called the product of this 
matrix by its transverse, will be the discriminant in question, or which is the 
same thing this discriminant is the sum of the squares of all the complete 
minors that are contained in the oblong matrix. 
In the case before us if we make /= x n — 1* it will easily be seen that 
* When f=x n -1 the value of S (the mean of the values of y) is obviously zero. Suppose 
now by way of illustration that n = 5, then calling the imaginary 5th roots of unity pi, p 2 , ps> pit 
one of the complete minors referred to in the text will be the determinant of the matrix 
Pi 
P2 
P3 
Pi 
Pi 2 
P2 2 
P3 2 
Pi 2 
Pi 3 
P2 3 
P3 3 
Pi 3 
Pi 4 
P2 i 
P3 4 
Pi*> 
and when the columns of this matrix are divided respectively by pi,p%, p$, pi, [0 = 1, 2, 3, 4], 
which will leave the value of the determinant unaltered, the determinant of the matrix so 
modified will represent in succession each of the other 4 minors. 
The value of the one above written, paying no attention to the algebraical sign, is by a well 
known theorem the product of the differences of p 1 , p 2 > P3> pit that is, inasmuch as 
(1-ft) (l-P3)(l-fc) = 5