IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 21 



variants, other invariants. One such process is differentiation: the 

 derivative, with regard to any one of the independent variables, of an 

 absolute invariant is, in its turn, an absolute invariant; for from 



1(a) = 1(a), 



dl(a) dl(a) 



follows. Since the quotient of any two 



dxi dxi 



relative invariants of the same degree is an absolute invariant, this proc- 

 ess supplies us with a means of deriving, from two such invariants, a 

 third; a result which, since the denominator, and therefore also the 

 numerator, of the derived invariant are themselves invariants, we may 

 state in another form as follows : If h, 1 2 be any two relative invariants 

 of the same degree, /*, then the Wronskian 



h 

 h 



dh 



dx 



dx 



is also an invariant, and is of degree 2fi. We note that this Wronskian 

 process admits of extensions. If, for instance, I\, 1 2, 1 3 be three invari- 

 ants of the same degree, /*, then both 



j v*2 



and 



h ^ 



are invariants. And in general the following precept may be laid down 

 for deriving invariants. Write down, as the first column of an ra-rowed 

 determinant, m invariants of the same degree, \i. Take for the elements 

 of any other column the derivatives, with regard to any given one of the 

 independent variables, of the elements of some preceding column. 

 This independent variable may be different for different columns. The 

 determinant so constructed will be an invariant of degree mp. The 

 proof consists in writing down the transformed Wronskian, when 

 everything except yjr m ' 1 times the original Wronskian will be seen to 

 vanish. 



In particular we may derive invariants by this Wronskian process from 

 our linear invariants, the b's. For instance, let b stand for any given 



