150 Proceedings of Royal Society of Ediriburgli. [sess. 
It has been shown for two cases, ^.e., for the eliminants of equations 
of the 3rd and 4th degrees, that the solutions are not illusory, 
because the diagonal contains a term of the form, which 
does not vanish. This proof may be generalised as follows : — 
Supposing the two equations of the degree transformed by the 
removal of all powers of y except the first, we may wwite, 
X + &c. } + {a 2 ny + = 0 
a? + &c. } + {b^nV + = 0 
+ . . . + 
+ &c. } + 
® + &c. } + { a^pc^ . . . + ^ 
x^ { h-^x^^ ■ ^ + &c. } + { h,pc^ . . . + ~ ^ 
ifcc. &c. &c. 
In the series of derived equations which are formed by the cross- 
multiplication of these pairs of equations (eliminating the quantities 
outside the brackets), we have from the eliminant of the first pair 
a term ; whence multiplying by y and we form two 
equations containing respectively the terms. 
Similarly from the eliminant of the second pair, after multiplying 
by y and 2 , we have two equations containing respectively the 
terms 
{a-^An+i)^'''~V \ and so on. 
Comparing these terms in order with the terms of an equation of 
the degree, n+1, arranged in the order given in the first page of 
this paper, we see that the constituents (t»i,& 2 «+i) of each successive 
line fall on successive columns ] the whole forming a diagonal 
which wants only the places of the first and last three lines and 
columns, and these are supplied, as in the example given, by the 
multiples of the original equations. Thus, the eliminant of the two 
complete n-ic equations contains a term and 
thus includes the unique term, Therefore, the eliminant 
does not vanish identically. 
Eliminants expressed in the form of determinants when formed 
according to the method above described, may be tested for errors 
in a very simple way, as follows : — 
I 
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