188 l.J Prof. J. C. Malet. On a Class of Invariants. 215 



But this equation is satisfied by U=0, HU=0; consequently the 

 equations 



aU + 6/3HU=0, H(*U+ 6/3HU) =0, 



are both satisfied by the relations 



U=0, HU=0. 

 Hence the theorem given in the text. 



VI. " On a Class of Invariants." By John C. Malet, M.A., Pro- 

 fessor of Mathematics, Queen's College, Cork. Communi- 

 cated by Professor Cayley, LL.D., F.R.S. Received 

 December 14, 1881. 



(Abstract.) 



This paper is concerned with two kinds of functions of the coeffi- 

 cients of Linear Differential Equations, which have certain invariant 

 properties. 



In the first part of the paper it is shown that every Linear Differen- 

 tial Equation possesses a certain number of functions of the coeffi- 

 cients which are unaltered by changing the dependent variable y to 

 yu where u is any given function of x, the independent variable. 

 These functions bear remarkable analogies to functions of the differ- 

 ences of the roots of ordinary algebraic equations, and many problems, 

 provided they involve only the ratios of the solutions of the differen- 

 tial equation, may be solved in terms of them ; for example, the 

 condition that two solutions y l and y 2 of a linear differential equation 

 of the third order should be connected by the relation y^ =y z a is 

 expressed in terms of two such functions of the coefficients of the 

 equation. This problem is analogous to that of finding the discrimi- 

 nant of an algebraic binary cubic. 



The second part of the paper is concerning functions of the coeffi- 

 cients of Linear Differential Equations which are unaltered by change 

 of the independent variable, and the theory of these functions is 

 applied to the solutions of problems involving only relations among 

 the solutions of the equation without the independent variable. 



In this part of the paper it is shown how to form the condition that 

 the three solutions y lt y 2 , y s of a linear differential equation of the 

 third order should be connected by the relation y\y%=y^-, which 

 relation, involving only ratios of the solutions, and not containing the 

 independent variable, can be expressed in terms of either class of the 

 functions of the coefficients considered in the paper ; these two 

 methods of writing the condition are accordingly given. 



vol. xxxrn. 



Q 



