17 



in the Transactions* of this Academy, by Professor Mac Cul- 

 lagh himself. 



As respects the reciprocal ellipsoid, of which the vector v, 

 in the equation lately marked (5), denotes a semidiameter, it 

 may be mentioned here that, with the same significations of 

 the symbols, the following equation holds good : 



(2/3S.a/3)^ = OS.^v)V(V.j3V.av)^ (8) 



with equations for other central surfaces of the second order, 

 regarded as reciprocals of central surfaces, which differ only in 

 the signs of their terms from this equation (8). The author 

 proposes, in a future continuation of the present communica- 

 tion, to illustrate this new form, as regards the processes of 

 obtaining and of interpreting it. Meanwhile he desires to 

 submit to the notice of the Academy the following construc- 

 tion, for generating a system of two reciprocal ellipsoids, 

 by means of a moving sphere, to which his own methods have 

 conducted him, although it may turn out to have been already 

 otherwise discovered. Let then a sphere of constant magni- 

 tude, with centre JE, move so that it always intersects two fixed 

 and mutually intersecting straight lines, AB, AB', in four 

 points, L, M, L', M', of which L and M are on AB, while 

 L' and M' are on AB' ; and let one diagonal LM', of the in- 

 scribed quadrilateral LMM'L', be constantly parallel to a third 

 fixed line AC, which will oblige the other diagonal ML' of 

 the same quadrilateral to move parallel to a fourth fixed line 

 AC. Let N be the point in which the diagonals intersect, 

 and draw AF equal and parallel to EN^; so that AENF is a 

 parallelogram : then the locus of the centre E of the moving 

 sphere is one ellipsoid, and the locus of the opposite corner F 



' See the beautiful paper entitled, "Geometrical Propositions applied to 

 the Wave Theory of Light. By James Mac Cullagh, F. T. C. D." Read 

 June 24, 1833. Transactions of the Royal Irish Academy, vol. xvii. 

 VOL. IV. C 



