458 ON THE ORDER OE CERTAIN SYSTEMS OF ALGEBRAICAL EQUATIONS. [77 
variables to be eliminated: the simplest example is the following : suppose a, b, a', b', a", b' r 
to be linear functions (without constant terms) of x, y, z, and write 
[ a£ + brj = 0, 
•< a'£ + b'v — 0, 
equations from which, by the elimination of £, 77, two relations may be obtained between 
the variables x, y, z. 
Suppose, however, from these three equations x, y, z are first eliminated : the ratio 
£ : 77 will evidently be determined by a cubic equation ; and assuming £ : 77 to be equal to 
one of the roots of this, any two of the three equations may be considered as implying 
the third ; and will likewise determine linearly the ratios x : y : z. Hence any deter 
minate function of these ratios depends on a cubic equation only, or the system is 
one of the third order. But the order of the system may be obtained by means of 
the equations resulting from the elimination of f, 77 ; and since this will explain the 
following more general example (in which the corresponding process is the only one 
which readily offers itself), it will be convenient to deduce the preceding result in this 
manner. Thus, performing the elimination, we have 
L = {alb" - a"b') = 0, L' = (a"b - ab") = 0, L" = (ah' - a'b) = 0. 
Here the equations L — 0, L' — 0, L" = 0, are each of them of the second order, 
and any two of them may be considered as implying the third. For we have 
identically, 
aL + a'L' + a"L" = 0, (*) 
so that L = 0, 11 = 0, gives a"L" = 0, or L" = 0. Nevertheless the system is imperfectly 
represented by means of two equations only. For instance, L = 0, L'— 0 do, of them 
selves, represent a system which is really of the fourth order. In fact, these equations 
are satisfied by a" = 0, b" = 0, (which is to be considered as forming a system of the 
first order), but these values do not satisfy the remaining equation L" = 0. In other 
words, the equations L — 0, 11 = 0 contain an extraneous system of the first order, and 
which is seen to be extraneous by means of the last equation L": the system required 
is the system of the third order which is common to the three equations L — 0, 
1! = 0, L" = 0. 
Suppose, more generally, that x, y, z are connected by p + 1 equations, involving p 
variables f, rj, £ , 
aÇ +bri + cÇ + ... = 0, 
a'i; + b'r) + c'Ç+ ... = 0 ; 
1 Also bL + b'L' + b"L" = 0: but since by the elimination of L", taking into account the actual values of L 
and L/, we obtain an identical equation, these two relations may be considered as equivalent to a single one.