Scheffers in the Mathematische Annalen, vol. xxxix., pp. 364–74, enunciates the following theorem:—If A is an algebra containing the quaternion algebra B as a subalgebra, and if A and B have the same modulus, A can be expressed in the form B C = A = C B, where C is a subalgebra of A every element of which is commutative with every element of B: in other words, if i1, i2, i3, i4 is a basis of B, it is possible to find an algebra C with the basis e1, e2, … ec, such that each of its elements is commutative with every element of B, and such that the elements eris(r = 1, 2, … c, S = 1, … 4) form a basis of A; and if a is the order of A, then a = 4c.