77] 
457 
77. 
ON THE ORDER OF CERTAIN SYSTEMS OF ALGEBRAICAL 
EQUATIONS. 
[From the Cambridge and Dublin Mathematical Journal, vol. iv. (1849), pp. 132—137.Q] 
Suppose the variables x, y ... so connected that any one of the ratios x : y : z,... 
or, more generally, any determinate function of these ratios, depends on an equation 
of the /A th order. The variables x, y, z ... are said to form a system of the y th order. 
In the case of two variables x, y, supposing that these are connected by an 
equation U = 0 (U being a homogeneous function of the order y) the variables form 
a system of the y th order; and, conversely, whenever the variables form a system of 
the y th order, they are connected by an equation of the above form. 
In the case of a greater number of variables, the question is one of much greater 
difficulty. Thus with three variables x, y, z; if y be resolvable into the factors y, y", 
then, supposing the variables to be connected by the equations U = 0, V = 0, TJ and V 
being homogeneous functions of the orders y, y", respectively, they will it is true 
form a system of the y th order, but the converse proposition does not hold: for instance, 
if y is a prime number, the only mode of forming a system of the y th order would 
on the above principle be to assume y = y, y" = 1, that is to suppose the variables 
connected by an equation of the y th order and a linear equation; but this is far 
from being the most general method of obtaining such a system. In fact, systems not 
belonging to the class in question may be obtained by the introduction of subsidiary 
1 This memoir was intended to appear at the same time with Mr Salmon’s “ Note on a Eesult of Elimi 
nation,” (Journal, vol. hi. p. 169) with which it is very much connected. 
C. 
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