294 
MR. R. S. HEATH OH THE DYNAMICS OF 
and therefore 
cos 6 cos B=aa'-\-bb'-\- cc'+ff'+gg'+hh'. 
19. Again, if we expand the determinant 
l m n p 
A. fJL V 77 
l' m' ri p' 
X' f V 7s' 
it becomes 
af' + bg '-fc h' -{-fa + g b'+be 
Now, squaring the determinant, we get 
A 3 = 1 . cos 9 
1 . cos 8 
cos 6 L 
cos 8 . 1 
= 1 — cos 2 6 — cos 2 8 + cos 2 9 cos 2 8 
= sin 2 9 sin 2 8. 
Hence 
sin 9 sin 8 —af'-\-bg'-\-cli-\-fa-\-gb'-\-hc. 
These formulae remain unchanged if we pass to the other common perpendicular, or 
if we take the two polar lines instead of the given lines. 
If the lines meet one another 
of + bg' + eh'-{-fa -{-gb' + he— 0, 
and then the angle between them is given by the equation 
cos 9=ad ' + bb' -t- cc f ff+gg' + hli. 
20. The equation to a sphere, whose centre is (/, to, n, p) and the cosine of whose 
radius is r, is 
lx + my + nz+pu = r. 
This may be written in the homogeneous form 
{ lx+my -f- nz-\-pu } 2 = r 2 (f + f 1 +z 2 +w 2 ) . 
Hence we infer that a sphere is a quadric touching the absolute quadric along its 
intersection with tire polar plane of the centre of the sphere. The polar plane itself is 
a particular case of a sphere, the radius being equal to a right angle. 
