In February 1939 the Gazette published a “diagram for the study and solution of triangles”, in which there was a one-to-one correspondence between each point within a certain area and a triangle of specified shape. The diagram which it is the purpose of this article to explain is even simpler in theory and is perhaps more effective for the solution of triangles, though it has not the interesting features of the one-to-one correspondence between a point and a triangle.

The following discussion of the curves r1r2 = crn, where r1, r2, r are the distances of a point from the “foci” ±α, 0 and the origin, c a positive constant and n rational (integral or fractional, positive or negative), provides a good example of extensive knowledge of general curves obtained by elementary-considerations. The attention of the author was first drawn to these curves by the remark of a colleague, Mr. N Davy, M.Sc., that the curves r1r2=cr⅓ appeared in the design of the pole-pieces of certain electromagnets.

Given a tetrahedron of reference ABCD, let λ, μ, ν, π be the cosines of the angles made by a line with the perpendiculars to the faces BCD, ACD, ABD, ABC respectively, these perpendiculars being drawn all inwards or all outwards; where

A, B, C, D being the areas of the faces BCD, .. If P1 (α1, ²1, γ1, δ1) and P2 (α2, β2, γ2, δ2) are any two points, the lines PP1 and PP2 can be written

to find the cosine of the angle between PP1 and PP2.