MR. SYLVESTER ON FORMULAS: CONNECTED WITH STURM’S THEOREM, 481 
two theorems is as follows. If a quadratic homogeneous function of any number 
of variables be (as it may be in an infinite variety of ways) transformed into a function 
of a new set of variables, linearly connected by real coefficients with the original set, 
in such a way that only positive and negative squares of the new variables appear in 
the transformed expression, the number of such positive and negative squares respect- 
ively will be constant for a given function whatever be the linear transformations 
employed. This evidently amounts to the proposition, that if we have 2w positive and 
negative squares of homogeneous real linear functions of n variables identically equal 
to zero, the number of positive squares and of negative squares must be equal to one 
another, so that ex. gr. we cannot have 
&c. ... &c. — mL} 
identically zero when n of the variables are linear functions of the remaining n ; and 
this is obviously the case, for if the equation could be identicallysatisfied we might make 
^a + 2 + 
and we should then be able to find as a real numerical multiple of and con- 
sequently should have the equation which is obviously impossible; 
a fortiori we may prove that in the identical equation existing between the sum of 
an even number of positive and of negative squares of real linear functions of half 
the number of independent variables, there cannot be more than a difference of two 
(as we have proved that there cannot be that difference) between the number of 
positive and negative squares. Hence there must be as many of one as of the other ; 
and as a consequence, the number of positive squares or of negative squares in 
the transform of a given quadratic function of any number of variables effected by 
any set of real linear substitutions is constant, being in fact some unknown trans- 
cendental function of the coefficients of the given function. I quote this law (which 
1 have enunciated before, but of which I for the first time publish the proof) under 
the name of the law of inertia for quadratic forms. 
Art. (45.). The other theorem is the following. If any quadratic function be repre- 
sented in the umbral notation* under the form of {ayXy-\-a^x.i-\- ... -\-a„x^‘^, where a^, 
a 2 .,.a„ are the umbrae of the coefficients, and Xy, X 2 ...x„ the variables, then by writing 
Xy-i- 
a. 
Oy 
Oy 
■3^2 + 
•3^2 + 
tty Ola 
.r-j- 
Qy Ua 
Uy Ug 
tty tt4 
^4-f ...+ 
x^-\- &c. -}- 
tty 
ttn 
tty Otg 
‘3'n — y \ 
Xn=yi 
tty «3 
Oy fltg 
0.2 
a2 
&c. 
^^2 ^^3 
^2 
— 3/3 
&c. &c. &c. 
* For an explanation of the umbral notation, see London 
or thereabouts. 
Cly OL2 
tty 
and Edinburgh Philosophical Magazine, April 1851, 
3 R 2 
