390 MR, A. B. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



20. But for our present purpose we require, not merely the expressions (12) for the 

 coefficients Q in terms of P, but also expressions for the derivatives of different orders 

 of the quantities Q. Writing (12) in the form 



so that we may, to the order of email quantities retained, differentiate <, with regard 

 to z or x indifferently, we have 



dsf ~ ' dsf C daf ' 

 But as before ( 1 L ) we have 



JrV m = r T} /7>P 



"> r i ^ -Pr.M <* * 



<fcf "~ m = i m! da?" ' 

 where 



B, t , 



. . e , f dx 1 , d*x 1 



= coefficient of P r in \PJ Z + 2iP*& + V. 



Now we have 



dx 1 

 - = j 



that is, 



cfa; 



3?" 77; 



to the first order of small quantities, and similarly for the differential coefficients ol 

 higher orders ; whence 



-^ = coefficient of p r in {p c (/D/A* + -j, p 2 ^ + ^, p 3 /i ui +)}" 



and, therefore, 



7T == l ~ 



while, for values of r greater than m, 



B r _ em 



rl r m + 1! dx r ~ m+l 



When these values are substituted, the equation for d r P,/dz r becomes 



d'V, "' = '-' r! d*P, d 



, _ , , 



' ~ r ^'~' 



dsf ' 'df m = , 7/1 1 ! r - m + 1! 



