849] ON THE INVARIANTS OF A LINEAR DIFFERENTIAL EQUATION. 
393 
We have thus 
p" — 3 (q' — 2pp') + 2 (r — Spq + 2p 2 ) 
as an “invariant” of the differential equation 
It is to be remarked that this is not what M. Halphen calls a “ differential 
invariant; ” he uses this expression not in regard to a differential equation, but in 
regard to a curve defined by an equation between the coordinates (x, y), and the 
differential invariant is a function of the derivatives y", y"',... which is either =0 
in virtue of the equation of the curve, or else, being put = 0, it determines certain 
singularities of the curve. Thus y" is a differential invariant; the equation y" = 0 
determines the points of inflexion. Again 
is a differential invariant, vanishing identically if the variables (x, y) are connected 
by any quadric equation whatever; it is thus the differential invariant of a conic. 
This last differential invariant is intimately connected with the above-mentioned 
invariant 
p" — 3 (q — 2pp') + 2 (r — Spq + 2p s ), 
viz. writing with M. Halphen 
we have 
p" — 3 (q' — 2pp) + 2 (r — Spq + 2p 3 ) = p" + 6pp' + 4p 3 
It is moreover noticed by him that, writing 
y", y"', y'"', y m "^a, 6b, 24c, 20d, 
respectively, the function in { } becomes 
= — (a~d — 3abc + 2b 3 ) ; 
the form under which he had previously obtained the differential invariant of the 
conic. As remarked by Sylvester, it is mentioned pp. 19 and 20 in Boole’s Differential 
Equations (Cambridge, 1859), that the general differential equation of a conic was 
obtained by Monge in the form 
9y"Y'" - 45y'Y'y"" + 40 (y"J = 0, 
which (representing the differential coefficients as just mentioned) becomes a 2 d—Sabc+2b 3 =0, 
but putting them = a, b, c, d respectively, it becomes 9a 2 d — 4>5abc + 406 3 = 0 ; the last- 
mentioned form presented itself to Sylvester in his theory of Reciprocants. 
C. XII. 
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