to the Cartesian Ovals in piano. 385 



ovals, circular cubics, &c), forming a distinct genus, and calling 

 for a special and detailed examination of their focal properties*. 



* In an evil hour for the cause of sound nomenclature, and by an over- 

 hasty generalization, a property of foci properly so called was substituted 

 for their true definition as centres of linear relationship. The temp- 

 tation was great, it must be allowed ; for the new definition gave a means 

 of describing each focus per se without reference to the associated points : 

 it calls to mind the analogous attempted definition of man as afeatherless 

 biped, and is open to the like kind of objections. The mischief being 

 done, and under cover of the authority of names so great that to root it 

 out seems now hopeless, the best remedy to apply is, as I think, that used 

 in the text, of distinguishing the centres of linear relationship as syzygetic 

 foci or foci proper. There is room for a grand chapter in the promised and 

 anxiously-expected new edition of Dr. Salmon's ' Higher Plane Curves/ 

 on a systematic and exhaustive development of the laws of foci proper, 

 and the algebraical philosophy, as it may well be termed, of true focal 

 curves, i. e. curves the distances of whose points from one or more sets 

 of fixed points are subject to linear relations. Nothing can be more 

 curious than the study of the way in which, starting from a given set of fixed 

 points, other foci (as in the Cartesian ovals) are found capable of replacing 

 one or more of the given ones, constituting the theory of substitution — and 

 then, again, how, as in the conic sections and in circular cubics, besides this 

 faculty of mutual substitutability of foci of the same set, one set may he 

 entirely replaced by one or more other sets, constituting the theory of plu- 

 rality or distribution. Algebra cannot but gain largely by these ideas of 

 substitution and distribution being fully worked out. 



In answer to my objections to the undue extension of the term focus, it 

 has been urged that a focus, as originally presenting itself in the theory of 

 conies, is susceptible of two distinct definitions — first as a member of a 

 syzygetic group, and again as a point whose squared distance from any 

 point in its curve is the square of a linear function of the coordinates, — that 

 it is legitimate to generalize the conception from either of these points of 

 view, and that the latter leads to the definition of a focus as a point whose 

 squared distance from any point in its curve, multiplied by a quantic, gives 

 rise to a second quantic containing a squared linear function as a factor. 

 But I answer to this, that the generalization is carried too far and too fast, 

 two steps in enlargement of the original condition being taken at once to 

 arrive at it; that the first step should be to define a focus as a point 

 such that the squared distance in question, multiplied by a quantic, viewed 

 as a function of the coordinates, shall be a perfect square ; and that when 

 this first step is taken, the foci so obtained are foci of a peculiar kind, and 

 probably retain their quality as foci proper, or centres of linear relationship. 

 At all events they possess the property of giving, by their junctions with 

 the circular points at infinity, multiple tangents to the curve, according to 

 the law stated in a previous foot-note concerning such foci. 



If the word focus is retained to signify the proper or syzygetic species, 

 some slight modification of the word may be used to denote the genus, viz. 

 foci which satisfy the larger definition of being points of intersection of the 

 simple tangents to the circular points at infinity. I thought of the word focal 

 for the purpose; but this is objectionable, for the reason that it would pro- 

 bably be found advisable to retain that word to denote the class of curves 

 which possess foci proper. On the whole, the word subfocus seems to me 

 best to meet the exigency of the case, and possesses the recommendation 

 of being capable (with dialectic variations) of passing current in each of the 

 five accepted tongues — Latin, German, French, English, and Italian, which 

 happily at the present day may be regarded as the common property and 

 inheritance of mathematical Europe. 



