392 
ON THE INVARIANTS OF A LINEAR DIFFERENTIAL EQUATION. 
[849 
We thence obtain the identities 
P a + 2P 2 - 3Q_p' + 2p 2 - 3q 
R* ~ r 3 
Pi + 3PR _r' + 3pr 
m i > 
P 3 r 11 
., . , *>' + 2» 2 — 3g , r' + 3pr 
so that we have - — and 
/^»3 
equation of the third order. 
as /3-seminvariants of the differential 
But these are by no means the best conclusions ; it is shown by M. Halphen in 
his great Memoir, “ Mémoire sur la reduction des equations différentielles linéaires aux 
formes intégrables,” Sav. Etrang. t. xxvm. (1884), pp. 1—297, see p. 127, that there 
is a function invariantive in regard to each of the two transformations, and which is 
thus an invariant, viz. this is the function 
p" — 3 (q — 2pp') + 2 (r — 3pq + 2p 3 ); 
this is for the first transformation unaltered, viz. it is 
= P" - 3 (Q' - 2PP') +2 (P - 3PQ + 2P 3 ), 
and for the second transformation it is only altered by the factor (p ~ 3 , viz. it is 
= </>r 3 {P 2 - 3 (Q 1 - 2PP X ) + 2 (P - 3PQ + 2P 3 )}. 
It is interesting to directly verify this last result. Performing the differentiations 
in regard to X, we find without difficulty 
P 2 -M. + + Ft* ~ ^ ^. 
Ql - 2PPl = ((Jl - Zppi) (p! 2 +P!(f)2 - 2p 2 4>l<t>-2 + 2 </</>! </>-2 + P<î>3 - ^ ’ 
P - 3PQ + 2Q 3 = (r - 3pg + 2p 3 ) <^ 3 + p<f> 3 - 3p 2 <^ 2 4- 3#^ 2 - 3 + Ÿs ; 
01 0i 
and thence 
P-2 - 3 (Q x - 2PP X ) + 2 (P - 3PQ + 2Q 3 ) = p 2 (p x -S(q 1 - 2pp x ) (pi 2 + 2(r-3pq + 2p 3 ) (p x 3 -p x <p 2 . 
But, introducing herein the derived functions in regard to x, we have 
Pi =P<pu ?i = q<Pi> P2 =p'4>2 + p"<pi 2 , 
whence 
P2<pi -p 1 <p2=p"<Pi s ; 
and the right-hand side becomes 
= (p x 3 [p” - 3 (q — 2pp') + 2 (r - 3pq + 2p 3 )}, 
which is the required result.