His younger sister Margaret, described him as a man often completely lost in his thoughts.
[5] Later they separated and she started working as a governess in Russia in 1907.
[6] Cox wrote four papers applying algebra to physics, and then turned to mathematics education with a book on arithmetic in 1885.
[c 1] Contracted to teach mathematics at Muir Central College, Cox became a resident of Allahabad, Uttar Pradesh from 1891 till his death in 1918.
[c 2][c 3][c 4][c 5] For instance, in his 1881 paper (which was published in two parts in 1881 and 1882)[c 2][c 3] he described homogeneous coordinates for hyperbolic geometry, now called Weierstrass coordinates of the hyperboloid model introduced by Wilhelm Killing (1879) and Henri Poincaré (1881)).
Like Poincaré in 1881, Cox wrote the general Lorentz transformations leaving invariant the quadratic form
He also formulated the Lorentz boost which he described as a transfer of the origin in the hyperbolic plane, on page 194: Similar formulas have been used by Gustav von Escherich in 1874, whom Cox mentions on page 186.
In his 1882/1883 paper,[c 4][c 5] which deals with non-Euclidean geometry, quaternions and exterior algebra, he provided the following formula describing a transfer of point P to point Q in the hyperbolic plane, on page 86 together with
On page 88, he identified all these cases as quaternion multiplications.
Subsequently, that paper was described by Alfred North Whitehead (1898) as follows:[7] Homersham Cox constructs a linear algebra [cf.
22] analogous to Clifford's Biquaternions which applies to Hyperbolic Geometry of two and three and higher dimensions.
He also points out the applicability of Grassmann's Inner Multiplication for the expression of the distance formulae both in Elliptic and Hyperbolic Space; and applies it to the metrical theory of systems of forces.
His whole paper is most suggestive.In 1891 Cox published a chain of theorems in Euclidean geometry of three dimensions: (i) In space of three dimensions take a point 0 through which pass sundry planes a, b, c, d, e,.... (ii) Each two planes intersect in a line through 0.
(iii) Three planes a, b, c, give three points bc, ac, ab.
Thus the planes a, b, c, abc, form a tetrahedron with vertices bc, ac, ab, 0.
In 1941 Richmond argued that Cox's chain was superior: H. S. M. Coxeter derived Clifford's theorem by exchanging the arbitrary point on a line ab with an arbitrary sphere about 0 which then intersects ab.
The planar language of Cox then translates to the circles of Clifford.
[9] In 1965 Cox's first three theorems were proven in Coxeter's textbook Introduction to Geometry.