Medium, according to the Principles of FresneL 5 



infinitesimals. To see this, it is only necessary to recollect that, 

 by the property of maxima and minima, the semiaxis differs 

 from any semidiameter indefinitely near to it, such as OQ, by an 

 indefinitely small quantity of the second order. Hence, since 

 OT is equal to OQ, and OT, to the above-mentioned semiaxis, 

 it follows that OT and OT, differ by an indefinitely small quan- 

 tity of the second order, and that therefore the angle OTT t is 

 ultimately a right angle : consequently the tangent to the curve 

 in which the plane TOq intersects the locus of T is perpendi- 

 cular to the plane PQOT. But the tangent plane at T passes 

 through this tangent, and therefore the perpendicular OS to the 

 tangent plane must lie in the plane PQOT. 



Again, let a point T f indefinitely near to T, and in the plane 

 PQOT, be taken in the surface which is the locus of T 7 , and let 

 the plane of the section which is perpendicular to OT' intersect 

 the plane PQOT in the straight line OQ' which meets the ellip- 

 soid in Q'. Then that semiaxis of the section to which T f is 

 equal will be indefinitely near to OQ', and will therefore differ 

 from it by an indefinitely small quantity of the second order. 

 Hence, since OT is equal to OQ, the angle OTT' will be ulti- 

 mately equal to OQQ' ; and therefore, TS and QP being tan- 

 gents, the angles OTS and OQP are equal. But OT= OQ, and 

 the angles P and 8 are right ; therefore 08 = OP, and the angle 

 SOT = POQ ; whence SOP = TOQ = a right angle. 



Similarly, if one perpendicular be let fall from on a plane 

 touching the locus in V 9 and another on the plane touching the 

 ellipsoid in <?, it may be proved that the two perpendiculars are 

 equal and at right angles to each other, and that, with the lines 

 OF and Oq, they lie in a plane at right angles to OQ. 



6. An ellipsoid being cut by any plane through its centre, 

 the difference between the squares of the reciprocals of the semi- 

 axes of the section is proportional to the rectangle under the 

 sines of the angles which the plane of the section makes with 

 the planes of the two circular sections of the ellipsoid. (See 

 Fresnel's Memoir, p. 150.) 



