1. Presented to the Society, Dec. 31, 1919.
2. Walsh, these Transactions, vol. 19 (191S), pp. 291–298. This paper will be referred to as I.
3. Maxime Bôcher, A problem in statics and its relation to certain algebraic invariants, Proceedings of the American Academy of Arts and Sciences, vol. 40 (1904), p. 469.
4. Bôcher’s proof (1. c, p. 476) is reproduced in I, p. 291.
5. The term envelope is used to denote the set of points which is the totality of positions assumed by each of the points z1, z2, z3, z4; the points z1, z2, z3 are supposed to vary independently. The proof of Theorem II which is presented in detail has some advantages and some disadvantages over the following suggested method of proof. The theorem is evidently true when C1, C2, and C3 are points. The theorem is easily proved when C1 and C2 are points but C3 is not a point. By taking the envelope of the circular region C4 in the preceding degenerate case, the theorem can be proved when C1 is a point but neither C2 nor C2, is a point. The envelope of the region C4 in this last degenerate case, as z1 is allowed to vary over a region C1 not a point, gives the envelope of z4 for the theorem in its generality. I have not been able to carry through the actual analytic determination of the envelope by this method because the algebraic work is too laborious. This suggested method of proof, however, shows at once that the boundary of the region C4 in the general case is an algebraic curve or at least part of an algebraic curve. It seems to me likely that Theorem II is true also when λ is imaginary, but I have not carried through the proof in detail. In general the relation of the regions C1, C2, C3, C4 is not reciprocal. For example if C1 is a point but neither C2 nor C3 is a point and if these regions lead to the fourth region C4, then if we choose the circular regions C2, C3, C4 as the original circular regions of the lemma, we cannot for any choice of λ be led to the region C1. This lack of reciprocality does not depend on the degeneracy of one of the regions C1, C2, C3, C4.