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



Differentiation of this gives 



= 2/*' 

 hence 



^ = IL "" 



V, /*" 



or, if we take a new variable s = /u, t , this may be written 



= 



(52). 



Now from the original equation we have at once 



x = az + B, 

 2 = 02 + C, 



and therefore 



B + az 



(52') ; 



this relation, in which A and B (and from the point of view of (52) a also) are 

 arbitrary constants, is the primitive of the equation (52). It evidently contains two 

 independent arbitrary constants. 



The linear derivative was s 1 , being connected with the linear equation du/dz = ; 

 the new derivative 



which is connected with the less simple form of linear equation, will be called the 

 hyperlinear derivative. 



121. Example II. For the equation of the second order 



we at once have, by double differentiation of the equation u. 2 = u^, the relation 



Successive differentiations of this and substitution for v}\ give 



= 3/t'a + 3 A'i + (/' 



= Q^a + ( V" - 2/t'Q.) ', + (^ - 



