Then if j, J be the original and transformed forms of an invariant J= (a1)**wj**, w being the weight of the invariant.

Operation upon J results as follows D AA J = **wJ**; D A J=0; D ï¿½A J =0;D ï¿½ï¿½ J = **wJ**.

Now D A xA k = (n - k) A k; Aï¿½ A k = k A?1; D ï¿½A A k = (n - k) A k+1;D mï¿½ A k = kA k; (n - k)A ka - w Ak - 1 aA k = O; a _ J (n - k) A k +l A k = O; kA k Ak = **wJ**; equations which are valid when X 1, X 2, ï¿½ 1, ï¿½2 have arbitrary values, and therefore when the values are such that J =j, A k =akï¿½ Hence °a-do +(n -1)71 (a2aa-+...

= **wj**, aa 1 aa 2 a a 3 the complete system of equations satisfied by an invariant.