On the Proposition that every Equation has a Root. 47 
wanting, let it be transformed, by adding a given quantity to its 
roots, nto an equation in which no coefficient is zero, as 
x +pat+qar+tra?tsa+t=0. 
Then supposing that 2=At+ Bi?+C#+ &c., it may be readily 
shown by the method of the reversion of series, that 
Ifit be supposed that #=a + bs + cs? + ds? + &c., the same method 
gives, 6, c, d, &c. by means of simple equations as functions of 
a; but a itself is given by the equation 
a + pat + qa? +ra*+t=0. 
From this equation a value of a may be obtained by the process 
just indicated, and thus 2 will be expressed in a series proceed- 
ing according to the powers of s. Similar reasoning applies to 
the other coefficients. 
These different series for 2 might be proper for finding real 
roots of equations ; but as they are not necessarily convergent, 
they do not prove that a root can always be found. They show, 
however, that w is an algebraic function of the coefficients; and 
as every algebraic function reduced to numbers is of the form 
z+y /—1, it may consequently be assumed that x is of that 
form. = 
3. Hence z+y V¥ —1 may be substituted for x in the given 
equation f(z) =0; and as after this substitution it does not cease 
to be an equation, we shall have 
fizet+y V—1)=0, 
or j, vl 
P+QvV—1=0, 
P and Q being real functions of z and y._ I am aware that ma- 
thematicians who have given especial attention to this question, 
have not thought themselves at liberty, after substituting 
z+y V —1 for 2, to equate the result to zero, but have endea- 
youred to prove by independent considerations that there are 
values of z and y which will make P and Q vanish simulta- 
neously. I confess that I am unable to see the necessity for 
this course of reasoning, which has the disadvantage of requiring 
a peculiar and complicated analysis, of the validity of which it is 
difficult to judge. It being once admitted, on the grounds 
above indicated, that the unknown quantity of an equation may 
have the form z+y “ —1, it must surely be also admitted that 
this expression may be put in the place of # without destroying 
the equation. According to the view that I take, the resulting 
