( 485 ) 
Mathematics. — ‘On Orthogonal Comitants”. By Prof. JAN 
DE VRIES. 
If we regard zj and #2 as the coordinates of any point P with 
respect to the rectangular axes OX, and OX», the binary form 
n 
== — --1 —2 2 
a; = (a, nj + ag 2) = ang dj +2 Gn—1,17, A2 (} ens, a i 
n 
ag Gon “2 
is represented by x» lines through 0, containing the points for which 
the form a” disappears. 
If 5, and & are the coordinates of P with respect to the rectan- 
gular axes O5} and O=.2, between the quantities 7}, 72 and 5, & 
exist relations of the form 
a = Ay, § + die Ss Ei = Ay mn + Agi Zon 
t= Aan 5 a hog Es 5e = Aig 2 + doo Tg. 
If by these substitutions the form a” is transformed into Rei 
have 
Az = A, 2 + ag 13 = (aj An + az Ao) Bie (a: Aye + Ag Ago) 52 
SO 
cr = Ay, ay + 9} a2, 
Gy = Ayg ay + hog ag. 
This proves that the symbolical coefficients aj, a2 and @,, «3 
are transformed into each other by the same substitution as the 
variables #j, zz and §, &. 
2. In order to obtain comitants, i.e. functions of a,, az, 2, za 
that are invariant with respect to the indicated orthogonal transfor- 
mations, we can start from the covariants 
Ha} aad (ay) + (%2—y2)*, 
representing the square respectively of OP and of the mutual dis- 
tance of two points P and Q, being therefore absolute comitants. 
The second covariant can be replaced by 
35 
Proceedings Royal Acad. Amsterdam. Vol. II. 
