Further Applications of Metrix Notation to Integration Problems

Gives the finite equations of the adjoint group of a continuous group with parameters not canonical in terms of " the matrix " of the first and second parameter groups. J. Gives the finite equations of the adjoint group with canonical parameters, showing that the exponential matrix A of a former note (Proe. Loud. Math. Soe. , Vol. xxxiv. , 1902, p. 91) enters therein; and so gives a simple proof of the so-called exponential theorem. 3. Remarks on the connection with Lie's formula;. 4. Shows that any transformation of the adjoint group can be resolved into a succession of two transformations respectively of the first and second parameter groups. 5. Shows that this leads to a result including as a particular case the theorem that the characteristic determinantal equation allows the adjoint group. G. Gives another proof of this theorem of invariance. 7. Develops the result further, establishing in particular the equation ip (c't, a, '. r) = A e >0. i) \ (x, eft). * The notation in the Eiuyk. Math. Wiss. , Vol. n. , A. G, p. 40G, seems an unfortunate change from Lie's for the matrix a in Transformationsgruppcn, Vol. I. , p. 'A4. If the quantity a Ba of the Encyk. be called y a/, then the matrix y is the negative of the matrix £ (a) belonging to the second parameter group.