AND ENVELOPES OF HOMOLOGY, 

 that is : — 



= V (fit - yZ) ./2 + m 2 ( y 2 _ a 2) , gfS + 7l 2 («,2 _ £8) _ #3 + ( (ft 2 _ ^2) ^y _ gh + ^2 _ J2) yct _ tyf + (J2 _ m 2) ^ ,y ? j-g"]. 



Multiplying [3] successively by/ #, A, we have 



= «../ 2 * * * + y.hf + p.fg . . [9]; 



/3.<7 2 + y.flF* + «./gr . . [10]; 



015 



= 

 = 



y.W + $.gh + a.hf 



[11]. 















7 



7 



P 







a 



a 







= 



Forming now the determinant of elimination from equations [1], [2], [8], [9], 

 [10], [11], we have, cleared of extraneous factors, the equation of the locus sought, 

 namely, 



I 2 l 2 ($ 2 -y 2 ) a 



m 2 m 2 (y 2 — a 2 ) 



n 2 n 2 (a 2 -B 2 ) 



« (m 2 -n 2 )l3 7 



/3 (n 2 -l 2 )y* 7 



7 (Z 2 -to 2 )«/3 /S a 



which is, when expanded, the very symmetrical curve, 

 /p+f- a y / y * + «'-ff y / g » + ffl-ys y /i i i\ / p + to» + »« \ 2 



V ^7 / V ™7« / ^ V ««£ / V 2 ™ 2 n 2 J \ Imn )■ 



It seems scarcely necessary to add that in equation [1] or (50), the curve in 

 which the inverse centre moves, is legitimately taken, rather than that in which 

 the centre itself moves, since in the latter case the other two equations change 

 places, and the same elimination has to be effected, the variables being merely 

 inverted. 



VOL. XXIV. PART III. 



8d 



