AND IMAGINARY ROOTS OR EQUATIONS. 
647 
When <p< — 3, <r is always positive, and proceeds continuously from oo to 0 as <p passes 
from — 3 — g (a being infinitesimal) to — oo. Consequently <r must not be allowed to 
have any positive value. When ® = oo, <r=0, and when <p= 1, <r= — f. 
Hence, if no minimum value of <r (i. e. no maximum value of — a) occurs between 
p=l, ®=go , o' may have any value between 0 and — f ; but if such a minimum value, 
— M, where M>f, should exist, the admissible values of <r would become more enlarged, 
and might be taken between 0 and — M. 
Making then 0, we have 
2 3f + 2<p - 6 
2<p + l <p 3 +f 2 — 6<p’ 
or 
4<p 3 +5p 2 -}-2<p— 6=0 ; 
which, substituting 1+^ for <p, becomes 
4^ 3 +17^ 2 +24^+5=0; 
so that there can be no real root of the equation in <p greater than unity. 
Hence the admissible values of cr are defined by the inequalities 0><r> — f, 
i. e. 0>- 1 -^>-f, or 0>— (l + g)> — 3, or 2> ? >- 1. 
(75) We have thus obtained the complete solution of the problem of assigning inva- 
riantive criteria, such that their signs (positive, negative, or zero) shall serve to fix the 
nature of the roots. These criteria we now see are 
J, D, A+^JD, 
where p (the negative, it must be noticed, of §) is any numerical quantity intermediate 
between 1 and — 2( 61 ). 
(76) This important modification of the original criteria J, I), A I proceed to apply 
to the problem of obtaining the simplest and most symmetrical expression for the criteria 
in terms of the roots of the equation. Let a , 5, c, d, e be the roots, and write 
Z =%{{a-b)\a-c)\b-c)\a-d)\a-e)\b-d)\l)-e)\c-d)\c~ey}, 
or say 
*-={«“. <* 2 1 )"- 
( 61 ) Strictly it has only been proved that the surface A + /xJD, which passes through the line A, D, contains 
no superior facultative points except those comprised in the line L=0, J=0. It is, I think, not difficult to see 
from this, that, if in the “sack” formed between A and A+jmJD any such points were contained, L=0, J=0, 
i. e. the axis of D would he a double or multiple line on the surface G, which is easily disproved by examining 
the algebraical form of G in art. 41, where K represents - ; any obscurity, however, which may be sup- 
posed to cling to this view is immaterial, as a demonstration capable of being followed in piano and leaving 
nothing to be desired in point of perspicuity, will be found in the Note appended to this Part. 
( 62 ) Agreeable to the meaning assigned to £ and to a couple of rows of letters in my memoir on Syzygetic Re- 
lations, in the Philosophical Transactions. 
