69 



fore differing from OQ by an indefinitely small quantity of the se- 

 cond order, adopting, for brevity, the language of infinitesimals. To 

 see this, it is only necessary to recollect, that by the property of 

 maxima and minima, the semiaxis differs from any semidiameter in- 

 definitely 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 quantity of the second order, and that therefore the 

 angle OTT, is ultimately a right angle : consequently the tangent to 

 the curve in which the plane TOq intersects the locus of T is perpen- 

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

 through this tangent, and therefore the perpendicular OS to the tan- 

 gent plane must lie in the plane PQOT. 



Again, let a point T' indefinitely near to T, and in the plane 

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

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

 PQOT in the straight line OQ' which meets the ellipsoid in Q'. Then 

 that semiaxis of the section to which OT' is equal, will be indefinitely 

 near to OQ' and will therefore diflfer from it by an indefinitely small 

 quantity of the second order. Hence, since OT is equal to OQ, the 

 angle OTT' will be ultimately equal to OQQ'; and therefore, TS 

 and QP being tangents, the angles OTS and OQP are equal. But 

 OT = OQ, and the angles P and S are right, therefore OS = OP, 

 and the angle SOT = POQ ; whence SOP ;=^ TOQ = a right 

 angle. 



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

 touching the locus in V, and another on the plane touching the ellip- 

 soid in q, it may be proved that the two perpendiculars are equal and 

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

 they lie in a plane at right angles to OQ. 



