40 
wherein A, B, . . , F are constants. Multiplying the first, 
second, and last of these equations into the arbitrary multi- 
pliers X, g, and v , and adding the results, we have 
(\A + /iB + vC)a + (XD + g E + FF)6 = Xa + yfi + vy (4) 
Hence if the ratios of any two of the quantities X, g, v to the 
third be so assigned as to satisfy the equations 
\A + /tB + rC — 0, 
XD + /iE + vF = 0, 
then the sinister of (4) will vanish independently of a and l, 
and the homogeneous linear relation 
Xa + /i/3 + vy -• 0 (5) 
will subsist among the roots a, /3, and y of the algebraical 
equation. When n~ 3 Ave (since the differential resolvent is 
homogeneous) have without reference to, but consistently 
with, the theorem, 
a + (1 + y — 0 (6) 
Combining the above theorem with one given by Abel and 
Galois, we conclude that: — 
If an algebraical equation have a differential resolvent 
of the second order, the algebraical equation is resoluble 
algebraically . 
Before closing I would add that, as it seems to me, it 
would be more consonant with the notation and practice of 
the rule of three, and, therefore, with convenience and the 
analogies of arithmetic, if by the ratio p : q there were univer- 
sally understood (not the fraction p + q, but) the fraction 
1+P- 
