Note on a Formula in Solid Geometry

To find the angle between the lines whose direction cosines are given by the equations

ul + vm + wn = 0, ... (I) and

al² + bm² + cn² + 2fmn + 2gnl + 2hlm = 0. ... (2)

The following variation of the method given in Bell's Three Dimensions. Art. 60, and Frost's Solid Geometry, Art. 26 may be interesting.