Lie's Theorem of the Torsion of a Complex Curve 69 



Consequently 



+ + di = 0, 



dp dg_ dg dq 



and therefore (in virtue of (B)) : 



dq dp 



On substituting the value of ^ given above in this equation, we find 



dp 



that the invariant ah + —c vanishes. 



Again let the assumed relation be 



^„ + i = S Ui{q)di. 

 From this relation and (B), we deduce 



; l£'./i!i+H)=o. 



>=i dq \ dp / 



If now we apply to (B) the Laplace transformation defined by 



e'=^ +b0. 



dp 



the transformed equation (B') will have n solutions 0/(i= 1, . . . . n) 

 connected by the relation 



Thus after w — 1 applications of this transformation, we shall arrive at an 

 equation having two solutions 5i^"7^^ ^2^""^^ connected by the relation 



and by the first part of the theorem, it has a vanishing invariant 



aA(»-l) 

 ^(n-l)^(«-l) + ^ _ c('-l). 



dq 

 Equations (12) and (17) lead to the relations 

 (18) a0i + 6^2 + c^3 =«a + 6,3 + C7 = 1, 



