IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 1 1 



slants, these conditions reduce to a { = 0. Thus Laplace's equation is 

 self-adjoint. 



For self-adjoint differential expressions, the S^'s in Lagrange's Iden- 

 tity reduce to 



_ ^ f du dv \ 



and that identity may be thrown into the form 



^c«)-2g=^>-2g. 



On the other hand we have for L(u), if self-adjoint, 



On inserting this value of L(u) in Lagrange's Identity above, the left 

 side goes over into 



2d -^ du " "^ d ^ du 



, ssL?' a "s?J + "-~?siL'f' H '^J' 



that is, 



2 3m dv 



Proposition 3. For self-adjoint differential expressions we get a 

 three-term form of Lagrange's Identity, 



T , N ^ dPi T . N s?dQi ^ du dv 



««•) - 2 s: = ■«•) - 2 j£ = - 2 -^ ^ + «■* 



the P's and Q's being given by (11). 



Integration would give a corresponding three-term form of Green's 

 Theorem. 



In conclusion, attention may be called to the fact that most of the 

 above can be made to apply directly (1) to ordinary differential ex- 

 pressions of the second order, (2) to differential expressions of the first 

 order, by simply putting the proper coefficients in L(u) equal to zero. 

 A similar remark is in order for the developments of the next paragraph. 

 We note that an expression of the first order can never be self-adjoint, 

 but may be the negative of its adjoint. 



