640 
PEOEESSOE SYLVESTEE ON THE EEAL 
(63) It remains only to fix the characters of the several regions; but this requires 
no calculation to effect, for we know that when D is negative there is one and only one 
pair of imaginary roots. This disposes of the first of the regions above enumerated. 
Again, we know that when L is positive so that the reduced form is the superlinear 
equation ri<!‘ -f si> 5 -}- tuf — 0 , u, v, w being real functions, D being also positive, there 
must be four imaginary roots, as follows from the theory of the second section. Hence 
the third region has for its character two pairs of imaginary roots ; and consequently 
the only remaining region, the second described, must correspond to the case of no 
imaginary roots, since otherwise we should be absurdly assuming the impossibility in 
any case of a quintic equation having all its roots real, 
(64) It may, however, be an additional satisfaction to see how the change of character 
comes to pass at the critical line OA from one to five real roots. 
Along the line OA we have found that, calling the reduced form ru*-\-svf-\-tw 5 , 
r=s -~=y=0+4=-4. 
q St t ' 
Hence the equation becomes 
4«q 5 + 4v* + (w y + vf = 0, 
u p v i being of the form — ^ W - •> because L is negative. 
Hence ^+^=0, or 
4(u* — n)v i -f u*vj — up] + v*) + (u r f ^) 4 =0, 
i. e. 5^+lOw^+v^O, 
i. e. 0; 
so that there are two pairs of equal roots of viz. +/ ; to these values of ^ correspond 
u—iv u — iv 
u + iv u-\-iv l ' 
Hence 
(1— l)v, or (1 +/)z^=(/ + l)v ; 
so that the two pairs of equal roots of | are +1, the outstanding root corresponding to 
u.-\-v ,= 0 being -=0. 
Now, still keeping upon the surface G, which we know is facultative, let & become 
— 8 -{-4s, where s is an infinitesimal, then 
=^=(40 3 +120 2 )^= —5120s ; 
also the supposed equation becomes 
( 4 - 4s )(u]+v)) + ( u, + vf = 0, 
or 
v (iv— uj— (w+w) 5 -f 8(l+s)w 5 =0; 
or, calling -=§■ 
(, ? -l) 5 -(, § +l)*+8(l+s)=0. 
