Let there be given, in a plane π, six (fundamental) points 1, 2, 3, 4, 5, 6, of which neither any three lie in a right line, nor all in a conic; and consider the six conics [1] ≡ 23456, [2] ≡ 13456, [3] ≡ 12456, [4] ≡ 12356, [5] ≡ 12346, [6] ≡ 12345, and the fifteen right lines 12, 13,…, 16, 23,…, 56.

There is a pencil of cubics 1223456 (curves of the third order, having a node at 1 and passing through the other fundamental points); their tangents at the common node form an involution, viz., they are harmonically conjugate with regard to two fixed rays. Five pairs of conjugate rays of this involution are already known; for instance, the line 12 and the conic [2] have conjugate directions at the point 1, for, they make up a cubic 1223456.

1. If in a circulant the places (r, s) and (p, q), owing to the cyclical permutation, be occupied by the same element, then the complementary minor of this element in its first place is to its complementary minor in the second as (−1)r+s: (−1)p+q.

To prove that the complementaries are as (−1)r+s:(−1)p+q is the same as to prove that the cofactors are identical. And as the cofactor of an element of a determinant is not altered by the transposition of rows and columns provided the determinant itself is not thereby altered, it is evident that all we have to show is that the element in the place (r, s) may by transposition of rows and columns be made to take the place (p, q), and the determinant remain in outward form the same as before. Now it is a known property of the circulant that this can be done by first bringing the element in the place (r, s) into the pth row by cyclical transposition of rows, and thereafter making an exact similar set of transpositions of columns. Thus the theorem is established.

The experiments which form the subject of this paper are designed, in the first place, to test the relation pointed out by Maxwell between Joule's discovery of the lengthening of iron in the direction of magnetisation, and Wiedemann's later researches into the twisting of iron under the influence of longitudinal and circular magnetisations, and, in the second place, to investigate the corresponding properties of nickel.

Let

be the equation to an algebraical curve of the nth degree, the co-ordinates of any point on which in a system of linear co-ordinates are (x, y, z), u0, u1, u2 …. being homogeneous functions of x and y of degrees indicated by the attached suffixes; then

is the equation to its Hessian, which is a curve of the 3(n − 2)th degree.

Every one of the 3n(n − 2) points of intersection of H and U is a point of inflexion on U if it be not a multiple point on U. In this last case the intersection may or may not be a point of inflexion on some one of the branches of U; but in any case where H passes through a multiple point the total number 3n(n − 2) of inflexions suffers a reduction.

The experiments contained in this memoir were made by the author with the galvanometer on wrought iron, steel, cast metal, &c, bars and plates, arranged in galvanic circuit in sea-water and various other waters and solutions, to determine the relative electro-chemical position of these metals under such circumstances.