404 MR. A. B. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT- 

 Since 6 = z'~ l , it follows that 



which can immediately be verified, [z, x] being the Schwarzian derivative 



Determination of the Linear Invariant of Index cr in its Canonical Form. 



31. The difference between the linear part of 6 ff (z) for a differential equation with 

 a non-evanescent Q 2 coefficient and the whole function 0,, (z) for the equation with an 

 evanescent Q. 2 coefficient lies, not in any difference between the two sets of numerical 

 coefficients, for passage from the former . to the latter is effected merely by making 

 Q 3 zero, but in the condition, that for the former & ff the independent variable z was 

 not determined and so could have an arbitrary value assigned, while for the latter this 

 independent variable is completely determined. In order, therefore, to obtain the 

 invariant Q^ in its canonical form it will be sufficient to determine the linear part of 

 6, in its uncanonical form, for which we adopt the same process as in the particular 

 cases 4 , 6 5 , 6 , Q 7 ; an arbitrary value is assigned to z, nearly equal to x, and the 

 coefficients of the linear part are determined, the remainder of the terms not being 

 necessary for a knowledge of the canonical form. To this we pass by retaining the 

 linear part alone, and the independent variable must then be considered determinate. 



We assume 



"~ 

 + ( l) a ~ 2 B,_ 2 _ 2 ii + a part which vanishes with Q. 2 . 



For the determination of the ratios of the constants B , B lf . . . , it is sufficient to 

 consider the terms involving /A" which occur when the invariantive equation 



is transformed by means of the equations (10), (12), (13). From the last two these 

 terms can at once be selected with the following results : 



(i.) In Q ff the term involving p." is 



_^ eo -( -_l) /i "P_ 1 ; 



(ii.) In dQ v _ l /dz the terms involving /t u are 



- e ( < r-l) A4 "P ff _ 1 -l (cr-l)(cr-2)^ < ^; , . - 



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



