77] ON THE ORDER OE CERTAIN SYSTEMS OF ALGEBRAICAL EQUATIONS. 459 
or what comes to the same by the equations (equivalent to two independent relations) 
a, a', 
a", . 
.. a,™ 
... 
b",. 
.. b { ® 
(where the number of horizontal rows is p). Consider x, y, z, as connected by the 
two equations 
a, . 
.. a № ~ 2) , 
= 0, 
a, . 
.. a ( P~ 2 \ 
ap 
b,. 
.. b [ P~ 2 \ 
b ( P~*> 
b,. 
.. b { v~ 2) , 
bP 
these form a system of the order p 2 , but they involve the extraneous system 
a, b, 
= 0. 
Suppose (j) (p) is the order of the system in question, then the order of this last 
system is (p — 1) and hence <£ (p) =p <i - 4> (P — 1) : observing that <£(2) = 3, this gives 
directly <f) (p) = \p (p +1). Hence the order of the system is %p(p+ 1). 
Suppose x, y, z, connected by equations of the form TJ — 0, V= 0, W = 0; U, V, W 
being linear in x, y, z, and homogeneous functions of the orders to, n, p respectively 
in rf y. By eliminating x, y, z, the ratio f : y will be determined by an equation of 
the order to + n + p; and since when this is known the ratios x : y : z are linearly 
determinable, we have m + n + p for the order y of the system. 
Thus, if m = n=p = 2, selecting the particular system 
arf + 2 btjy + cy 2 = 0, 
brf + 2ci;y + dyf = 0, 
erf + 2d£y + erf = 0, 
it is possible in this case to obtain two resulting equations of the orders two and 
three respectively, and which consequently constitute the system of the sixth order 
without containing any extraneous system. In fact, from the identical equation 
• (arf + 2b%y + erf) 
— (4d£ + 2ey) (brf + 2c%y 4- dy 2 ) 
+ (3c| + %dy) (erf + 2d%y + erf) 
= (ae - 4<bd + 3c 2 ) rf, 
58—2