474 MR. A. E. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 

 SECTION VIII. 



CHARACTERISTIC EQUATIONS SATISFIED BY CANONICAL FORMS or INVARIANTS 



AND COVARIANTS. 



Reproduction of Canonical Form. 

 124. When the differential equation of order n is taken in its canonical form 



d*u n\ _ d*~ s u 



d* ~*~ Sln-Sl^d*^*' 



and is transformed so as to have a new dependent variable rj and a new independent 

 variable then, from the investigation in 12, it follows that, if we take 



the new equation will be without the term in d*~ l ri/dl; a ~ } ; and, from the investigation 

 in 30, it follows that, if be determined by the equation 



{ z] = o, 



the new equation will be without the term in d*~ 2 ^/c?^"~ 2 , that is, the new equation is 

 in its canonical form. The last equation gives 



t az + b . . 



^=^M ' '''" - ' < 56 )> 



where a, 6, c, d are the constants ; and the equations (55) and (56) give the relations 

 by which a canonical form of differential equation can be transformed into a canonical 

 form. 



125. As we are proceeding to investigate, by the method of infinitesimal variation, 

 the partial differential equations which are satisfied by the concomitants in their normal 

 forms, it will be convenient to adopt the process of 19 and make nearly equal to z. 

 Thus, taking in (56) the determining conditions b = 0, a = d, c = \ ed, where c is 

 infinitesimal so that its square may be neglected, we have 



r = c, 



