KKIUVATIVKS ASSOCIATED WITH LINEAR DIFFKKKNTI \ L EQUATIONS. 407 



so that B,(z) is independent of n ; and therefore any invariant of an equation in its 

 canonical form is also an invariant of all equations of higher orders when in their 

 canonical forms. 



SECTION III. 

 DERIVED INVARIANTS OF A LINEAR DIFFERENTIAL EQUATION. 



Quadrinvariant*. 



33. We are now in a position to construct an infinite number of invariants which 

 are linearly independent (and also algebraically independent) of one another ; they 

 are, in the first instance, functionally derived from the n 2 invariants obtained in 

 the last section, which may, therefore, be called the fundamental invariants or the 

 priminvariants of the equation. The method of obtaining the first set of n 2 new 

 invariants is that adopted by BRIOSCHI for the cubic. 



With the notation already adopted, we have for the general priminvariant 



and thence by (8), after taking logarithmic differentials, 



and, therefore, also 



<rZ'= (log e.(x)} - z*' {log e,(*M - *' 2 {log .(*)! - 



Substituting in (6) and writing 



*. (*) = 2o- {log 9, (x) } - [ {log 8, (x)} J-^ P z , 



we have 



*"<!>, (2) = *,(*), 



80 that <!> is a new invariant with index 2 derived from the priminvariant 0, ; and 

 from every priminvariant such an invariant can be derived. Now, when the equation 

 is taken in the canonical form, the value of *, (z) is 



*. = 2<r (lo g e c ( 2 )} - [^ {lo g e.(z)}J 



