66 



soid, and a, /3, y, the angles which they make with a perpendicular 

 from the centre on a tangent plane, the square of the perpendicular 

 will be equal to 



a^ COS.- 06+^' •^os-' /3 + c' COS.' y. 



Let a plane through the point of contact Q 

 and one of the semiaxes OA, intersect the ellip- 

 soid in the ellipse AON, and the tangent plane in 

 the tangent QL, and draw QM perpendicular to 

 OA ; then OA is a semiaxis of the ellipse AON, 

 and therefore is a mean proportional between 



OM and OL ; whence OL = -j > denoting OM by x. But if p de- 

 note the length of the perpendicular from the centre O on the tangent 

 plane at Q, the cosine of the angle which it makes with OA will be 



equal to ol' and therefore 



Similarly 



Hence 



px , px 



COS. « = -S- , and a cos. « = s— 



a' a 



b COS. ^ r: -^=^ , and c cos. y z= J-— . 



c 



«"- cos.^ « + 6^ cos.^ ^ + c^ cos.^ y = / ( ^ + £ + £.j = p=. 



Cor. Since a', y, z, are as the cosines of the angles which OQ 

 makes with the semiaxes, it appears from the demonstration that the 

 cosines of the angles which the perpendicular to a tangent plane 

 makes with the semiaxes are, with respect to each other, directly as 

 the cosines of the angles which the semidiameter through the point 

 of contact makes with the semiaxes, and inversely as the squares of 

 the semiaxes themselves. 



2. If the semiaxes a, b, c, and a', b', c', of two concentric ellip- 

 soids, coincide in direction and be reciprocally proportional, so that 



