646 
PROEESSOR SYLVESTER ON THE REAL 
whose equation is A=0, D=0, and obviously is the only invariant not exceeding the 
twelfth order capable of so doing ; it only remains to ascertain within what limits the 
numerical coefficient g> must be taken so as to fulfil the condition that the combined equa- 
tions A— g>JD=0, G=0 shall be incapable of being satisfied by any positive value of D. 
(72) Substituting for A and D their values, the equation to be combined with G=0 
becomes 
J 3 — 2 u L+gJ(J 2 — 128K)=0. 
Returning to the notation of art. (55), and dividing by JK, this equation, when G=0, 
becomes 
<?-2"f+f(2-128)=0, 
or 
(l+e)qv-2 u q=128gv, 
which, substituting for q, v in terms of 6, gives 
or 
(l + § )0 5 (0+4) 2 ni 
0 + 6 — ^ 
0 3 + 40 2 
0 + 6 
128^ 3 (0+4-), 
(fl+4)0 2 (3 + 8)((3 3 -43 2 +323-256)+(3 3 -43 2 -960)g)=O. 
When 5 -f- 8= 0, D=0, see art. (57); neglecting, then, this factor, the condition to be 
satisfied is that when from the equation 
(3+4)3 2 ((3 3 -43 2 +323-256)g+(3 3 -4S 2 -963)) = 0 
a value of 3 has been deduced, the values of D corresponding thereto shall not be a 
positive finite quantity. 
(73) Now 
D_ i 128(0 + 6) 0 3 + 40 2 — 128(0 + 6) (0 + 8) 2 (0-12) 
J2~ i— 0 2 (0 + 4) 0 2 (0 +4) — 0 2 (0 + 4) 
If 0=0, or 3+4=0, since D cannot be infinite, we have J=0, so that A— gJD be* 
comes identical with the original criterion A. Hence the factor (3 + 4)5 2 in the quantity 
just above equated to zero may be neglected, and the condition to be fulfilled by g is that 
if 3 be any root of the equation 
— 0 3 + 40 2 — 320 + 256 
0 3 — 40 2 — 960 
3 shall be between — 4 and 12 ; this equation on making 3= — 4<p, so that l>p>— 3, 
becomes 
_ <? 3 + p 2 + 2<? + 4 
_ j ^ <p 3 + <p 2 — 6<p 5 
or, writing a = — 
2<p + 1 2<p + 1 
<r— f + f— 6f“(«p-2)f(<p + 3)’ 
(74) We wish to ascertain what values of a will be incompatible with the violation of 
the limits just assigned to <p, and accordingly we must inquire what is the range of values 
assumed by <r when <p > 1 or <p < — 3 ; any values of <r not included within this range will 
be admissible for the purpose in view. 
