594 EEV. HUGH MAETIN ON CENTRES, FAISCEAUX, 



The proof of this is, mutatis mutandis, the same as before. And generally,— 



III. If the centre of homology move in the curve 



i i i 



±(ldf ±(m/3j' T ± (n 7 )" = . . . . (14), 



the linear faisceau of homology will envelope the curve, 



i i i 



± {lu) n+1 ± (mp)^ 1 ± (ny)" + 1 = . . . . (15). 



IV. If the centre of homology move in a conic with respect to which the 

 original triangle is self-conjugate, namely, 



l-c? + m' 2 /3 2 + n 2 7 2 = (16), 



the linear faisceau of homology will envelope the curve, 



(laf + (mft* + (nyf = (17). 



The proof of this is the same as before ; and it follows also from III., n being 

 taken equal to ^. 



Of course, in (16), that the conic may not be imaginary, one or other of the 

 terms /, m, n, must be affected with the coefficient s/{ — 1); but the equation is 

 maintained in this form for the sake of symmetry. 



V. If the centre of homology move in a conic circumscribing the original 

 triangle, the linear faisceau of homology will be a faisceau pivotante. 



The equation of the circumscribing conic is 



I + m « . =Q (18) _ 



" p y 



Replacing a, (3, y, by the co-ordinates of the centre, we have, 



If + mg + nh = (19). 



And, as before, /« + g& + h = = « • • • 20). 



And the result is that 



a : j3 : y : : I : m : n (21). 



That is, the line of homology revolves round the fixed point I, m, n. This is a case 

 of what the French writers call courbes pivotantes,* the curve here being of the 

 first degree, the straight line. The number of pivots about which a faisceau of 

 curves of the degree s revolves, or through which every member of the faisceau 

 piwtant passes, is of course s 1 . We shall meet with a case of this subsequently, 

 in reference to curves of the second degree — conic sections. 



VI. If the centre of homology move in the circle circumscribing the original 



* The reader will find this subject elegantly treated in " Etudes Analytiques sur la Theorie General 

 drs Courbes Planes," par M. Felix Lucas (Paris, 1864); where, among other results, a very pretty 

 proposition concerning a property of conies circumscribing the same quadrilateral (due to M. Lame), 

 is thus generalised : — Lcs polaires d'ordre quelconque d'un point du plan, relativement aux diverses 

 courbes d'un faisceau pivotaut, foi'ment elles-mSmes vn faisceau pivotant. 



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