592 REV. HUGH MARTIN ON CENTRES, FAISCEAUX, 



curve that may be found ; and the inverse line of homology, displacing itself, will 

 generate another faisceau, whose envelope also may be sought. Farther, the lines 

 of homology, the direct and inverse, will in general intersect, and the point of 

 intersection, partaking of the motion of the system, will describe a locus, which 

 may also be made the subject of inquiry. The points in which perpendiculars 

 from the centres of homology meet their respective, direct, and inverse lines of 

 homology will also present certain loci when the centre of homology describes 

 any. given curve, and the perpendiculars themselves will generate faisceaux. 

 whose envelopes may be sought. 



Farther; this system of problems may be generalised and greatly extended. 

 The line, called the line of homology, has originated in a particular geome- 

 trical consideration ; but it may be considered as unshackled from its geometrical 

 genesis. The conception may be idealised, and conceived of as not restricted to 

 the straight line, the curve of the first degree, but as a curve of any degree what- 

 ever. It may be a function of any degree in the variables, and of any degree in 

 the co-ordinates of the centre as parameters ; and we may thus have faisceaux 

 of curves of homology of any order, and envelopes corresponding to them as 

 before. Moreover, the general problem may be inverted, and instead of inquir- 

 ing, " What envelope will the faisceau of homology generate when the centre 

 of homology moves in a given curve?" we may inquire, " In what curve must 

 the centre of homology move, in order, with a given curve of homology, to beget 

 a given envelope?"* Or the inquiry may take yet another direction, "What 

 must be the form of the curve of homology in order that its faisceau, generated 

 by the centre of homology moving in a given curve, may produce a given enve- 

 lope ?" Some instances of the problem, in all these forms, will be found in the 

 following pages. It is evident that the general inquiry may be prosecuted in 

 such directions as would task the uttermost resources of modern discoveries in 

 the theory of linear transformation and of canonical forms. With the exception 

 of an instance of elimination by the aid of the Jacobian and its differential co- 

 efficients, we shall not pursue it to the necessity of laborious calculations ; and 

 shall, for the most part, restrict ourselves to the conic sections, and to such forms 

 of these as are most manageable, and do not demand intricate elimination. These 

 forms are, — First, the conic circumscribing the triangle of reference, and of the 

 general form, 



-+«- + - = ° W; 



a P 7 



Second, the conic touching the three sides of the triangle, viz., 



d= (uaf ± { v $f ± (w 7 y = . . . . (2); 



* The idea of the inverse of the problem of the Envelope seems first to have occurred to Boole. 

 See his paper — characterised by his usual high generality and beautiful originality — in " Cambridge 

 and Dublin Mathematical Journal," vol. vii. p. 156. 



