24 Mr. Mac Cuttacu on the dynamical Theory of 
these diameters will be the axes of the ellipse in which their plane intersects the 
ellipsoid. 
For, the above condition expresses that either diameter is parallel to the tan- 
gent plane at the extremity of the other; they are therefore conjugate diameters 
of the elliptic section, and hence, as they are at right angles to each other, they 
must be its axes. 
Je * the equation of 
If the semiaxes of the ellipsoid be represented by * b 
condition will become 
a’ cos a cos a’ + 0° cos B cos fp’ ++ c* cosy cosy’ = 0. (a’) 
Lemma IV. Let s, s’ be the lengths of perpendiculars let fall from the 
centre of an ellipsoid upon any two tangent planes, and 7, 7” the lengths of radii 
drawn to the respective points of contact. Then putting w for the angle between 
the directions of r and s’, and w’ for the angle between the directions of 7’ and s, 
we shall have 
Ts COS wW =7" Ss’ COS w’. 
For if the semiaxes of the ellipsoid, having their lengths denoted by a, 4, c, 
make with the direction of s the angles a, B, y, and with that of s’ the angles 
a’, 6, 7/3 with the direction of r the angles a,, By yY» and with that of r’ the 
angles a, 8, y,; there will exist the relations 
@ COS a = TS COS a, b’ cos B = rs cos By, CG” COS ¥ = TS COS Yu 
a’ cos a = 7's’ cos a, b cos B’ = 7’ s’ cos B, ce cos 7 = 7's’ cosy, 
by one set of which the quantity 
a’ cos a cos a’ + b° cos B cos B’ + c? cos y cos %/ 
will be converted into 
rs (cos a, cos a’ +- cos f, cos 6’ + cosy, cos 7) = rs cos w, 
and by the other set into 
7 s’ (cos a, cos a + cos B, cos 6 + cos ¥, cosy) = 1’ 8’ COs w’ ; 
so that we shall get 
7s cos w = 7" s' cos w' = a cosa cos a’ + b cos B cos B’ + c? cosy cos 7’. (4) 
