41 
Prof. Sylvester on Derivation of Coexistence, 
the index of periodicity (?z), be the same as the number of re- 
peated terms {x y z ... /), then one relation exists between the 
coefficients: this is found by making the (?^ — 1) new bases 
coincide with [n — \) out of the old bases. We get accord- 
ingly, as the result of elimination, 
f PD {o a h c ,,, T) =. 0. 
Art. (11.) Cor. (2.) Let the number of equations be one 
more than that of the given bases, there will then be two equa- 
tions of condition. These are represented by preserving one 
new arbitrary base, as A. The result of elimination being in 
this case 
f PD [o a h c ,,, I K) =0. 
Ex. The result of eliminating between 
, X , y = 0 
a^, X + b^.y — 0 
«3 . jr + Z>3 . y = 0 
is ^ PD (o « ^ A) = 0 
i, e, A3 . ciy — A3 . + Aj . Z>3 “ Aj . ^2 + ^2 • ^3 
— A2 . Z>3 «! = 0, 
from which we infer, seeing that A3 A2 \ are independent, 
b^ . fq — . «2 = 0 
Z>3 . «2 “ ^2 • ^3 = ^ 
. ffs - *3 . = 0, 
any t*wo of which imply the third. 
In like manner, in general, if the number of equations 
exceed in any manner the number of bases or repeated 
terms, the rule is to introduce so many nenx) and arbitrari/ 
bases as together wiih the old bases shall make up the num- 
ber of equations, and then equate the zeta-ic product of the 
differences of zero, the old bases and the new bases, to nothing. 
Art. (12.) Cor. (3). Let the number of equations be 07ie 
fewer than the number {n) of bases or repeated terms; the 
number of introduced bases in the general theorem is here 
{?i — 2). Make these (?i — 2) bases equal severally to the 
bases which in the type equation are affixed to z, u,,,t, 
then C = 0 
D = 0 
L = 0, 
and we have left simply 
f PD {o a c d ,,, k 1) X ^ PD {obc d ... kl) y — 0. 
In like manner we may make to vanish all but A and C, and 
thus get 
^ PD {0 ab d ,,^kl) X ? PD {pcbd kl) z = 0, 
