If we make x = sin p, and y = sin p1, we get

This agrees with Gauss's transformation.

To deduce that of Landen from it, we change into I, and y into y, in the equation

A SIMPLE METHOD OF DETERMINING THE ANHARMONIC FUNCTION OF A CUBIC.

By WILLIAM S.

M'CAY, A. M., Fellow of Trinity College, Dublin.

=

1o. "THE several values of the anharmonic ratio determined at any point of the conic + y2+ 2a o by its intersections with the conic ax2 + by2 + cz2 = o are the ratios of differences of a, b, c." For this anharmonic ratio is determined at an intersection of the conics by the tangent to the first, and the lines to the vertices of the triangle of reference. At the intersection whose co-ordinates are

the tangent is

√b-c1 √c-a, Va-b,

x√ b−c+y√ c−a + z√ a − b = 0,

and one of the other lines is

So the anharmonic ratio in question is determined on z by these two lines and the vertices zx, zy, and therefore is

a - C

b-c

And generally the anharmonic ratio determined at any point of a conic (C), by its intersections with another (C'), is a ratio of differences of the roots of the discriminating equation

λ Δ + λ θ + λθ' + Δ' = 0.

2o. "The points of contact of tangents to a cubic from a point on it are the intersections of the polar conics of the

point with regard to the cubic and its Hessian" (Salmon's Higher Plane Curves, Second Edition, p. 199).

This is an immediate consequence of the theorem that the tangent to a cubic at a point A meets the cubic again at a point B, determined by the polar line of A with regard to the Hessian; fixing B, A is seen to be an intersection of the conics in question.

are

Using the cannonical form, the Cubic and its Hessian

The form of H shows that the polar conic with regard to the triangle of reference passes through the intersections of the polar conics with regard to U and H.

I shall apply the first theorem to find the anharmonic ratio determined at any point of the polar conic (C), with regard to the cubic by the polar conic (C'), with regard to the triangle of reference of a point on the cubic, this ratio being the required Anharmonic Function of the cubic.

The polar conics are

C = x2x2 + y'y2 + z'z2 + zm (x'yz + y'xz + x'xy) = 0.

C' = 2 (x'yz + y'xz + z'xy) = 0.

Forming the invariants of C and

Δ

C, we find

Ꮎ -
→ = 18 m2 x'y' z'.

A' = 2x'y' z'.

And when U' = o, the discriminating equation becomes

==

d3 (1 + 8m3) + 18 \2 m2 + 1 2 \ m + 2 = 0.

The disappearance of the co-ordinates already indicates the constancy of the anharmonic ratio of four tangents as their intersection moves along the cubic, as first pointed out by Dr. Salmon (Higher Plane Curves, Second Edi

tion, p. 142). Removing the second term of this equation (which does not alter the differences of the roots), putting λ (1 + 8 m3) = t, and writing S, T for m' – m, 1- 20 m3 – 8m3, the equation becomes

t3 - 12 St + 2 T = 0.

The equation which gives the differences of the roots of this form is well known to be

Reducing the roots of this by the factor √3S, it becomes

The ratios of whose roots are the values of the Anharmonic Function of the cubic.

This is the same equation that Dr. Salmon derives from other considerations (Higher Plane Curves, p. 192).