410 MR. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT- 

 SECTION VI. 



APPLICATION TO EQUATIONS OF LOWEST ORDERS. 

 Equation of the Second Order. 



80. For the equation of the second order there are no invariants. So far as 

 concerns the reduction of the equation to a normal form, it is at once evident that, by 

 a literal application of the result in 30, the equation would be reduced to the form 



by the solution of a linear equation of the second order. There is thus no simplifica- 

 tion or advantage in the reduction, for the original equation of the second order might 

 as well be solved, the subsidiary equation being, in fact, identical with the original:* 



But it is interesting to notice bow the well-known theory of the solution of the 

 equation of the second order is contained in the general results. In the case of 

 n = 2, we have, by (iv.), 



By the transformation y = \u the equation 



is transformed to 



provided (21) z be determined by the equation 



{*, x] = 2P 2 . 



The two independent solutions of the transformed equation may be taken to be 

 1 and zf; and hence the two solutions of the y-equation are 2'"* and 22'"*. And z is 

 now the quotient of two solutions of the original equation.t 



f}. ,.',,.;.>, !_. vim :> ;;.,f 3,,.,' ,/i.tZv ;.. ".v ^.O 



* This result may be compared with the result of applying TSCHIRNHAUNEN'S transformation to the 

 general algebraical quadratic equation. 

 f See my ' Differential Equations,' p. 92. 



