3S PROCEEDINGS OF THE AMERICAN ACADEMY. 



With the help of these four equations, I = %(h + ^2) reduces to 



I= n(n— l) f f top+2.g ■ da P +hq+i \ ^ f da p+ i, Q+ 1 da p ,g +2 \~\ 

 2 |_\ dx dy J 2 \ dx dy J J 



w — 1 f n - 1 V da P +\, q da p , q+ i ~\ 



+ —£- [Kiap+i* + «2Wi j - ~2~ [_-^- + — - j + a pq , 



p + q = n — 2. 



For ordinary differential expressions this reduces, as it should, to 

 the invariant of the differential equation which we have called, (23), 

 page 29, A n _ 2 /6 or I n -2, if we put, as proper, 



*i = — 



a n — 1 

 na n 



*2 = 0. 



For the second order, n = 2, m variables, the corresponding- inva- 

 riant is: 



where ^4j, is the cof actor of aij in 



Oil 



ai, 



dml • • • &mm 



This becomes for two variables, m = 2, 



= 27T 4cl4 — (ai 2 a22 — 2 a\a 2 a\ 2 + Gt2 2 aii) 



ay 



,'dan dai 2 \ 

 2(a 1 a 22 --a 2 a 12 )[ — + — j 



+ 



„, N /3ai2 . 3a 22 \ . f da x da 2 \ 



yl = ana 2 2 — ai 2 2 . 



(0 



Invariants of a partial differential expression analogous to A n —k/6. 

 We have found now invariants of a partial differential equation analo- 

 gous to the invariants A n ^ k /6 of an ordinary differential equation. It 



(i) 



remains to discover the analogue of A n —k/0, which, we remember, 

 was an invariant of the differential expression. This merely amounts 



