120 jMr. J. J. Sylvester's Enumeration of the 



tlic theory of the contacts of loci not transcending the limits of 

 \ailgar space^ by which I mean the space cognizable through the 

 senses *, and shall accordingly be almost exclusively concerned 

 in determining the singular cases of conjugate systems of qua- 

 di'atic forms of two, three, and four letters respectively. 



In order that the reduction of any such system, say U and V, 

 to a pure quadratic form may be possible (as it generally is), it 

 is necessary that none of the roots of the complete determinant 

 of U+XV shall be equal; if any relation of equality exist be- 

 tween these roots, the general reduction is generalhj no longer 

 possible; under peculiar conditions, however, as will hereafter 

 appear, in spite of the equality of certain of the roots, the irre- 

 ducibility in its turn will cease, and the ordinary reduction be 

 capable of being effected. It is easily seen, that to every relation 

 of equality bet^^en the roots of the determinant of U + A.V must 

 correspond a particular species of contact between the loci which 

 U and V characterize. But we should make a great mistake 

 were Ave to suppose that every such relation of equality corre- 

 sponded with but one species of contact ; for instance, the cha- 

 racteristics of U and V of two conies are functions of three let- 

 ters, and I I (U + A,V) will be a cubic function of X. Such a 

 function may have two roots, or all its roots equal : this would 

 seem to give but two species of contact, whereas we well know 

 that there are no less than four species of contact possible be- 

 tween two conies. Accordingly we shall find, that, in order to 

 determine the distinctive characters of each species of contact, we 

 must look beyond the complete determinant, and examine into 

 the relations (in themselves and to one another) of the several 

 systems of minor determinants that can be formed from U + W. 



By pursuing this method, we may assign a p-iori all the 

 possible species of contact between any two loci of the second 

 degi-ee. How important this method is will be apparent fi-om 

 the fact, that not only have the distinctive characters of the 

 various contacts possible between surfaces of the second order 

 never been determined, but their number and the nature of cer- 



* If the impressions of outward objects came only through the sight, and 

 there were no sense of touch or resistance, woukl not space of three dimen- 

 sions have been physically inconceivable ? The geometry of three dimen- 

 sions in ordinaiy parlance would then have been called ti-anscendcntal. But 

 in veiy truth the distinction is vain and futile. Geometry, to be properly 

 imderstood, must be studied under a universal jioint of view ; every (even 

 the most elementary projiosition) must lie rcgartled as a fact, and but as a 

 single specimen of an infinite series of homologous facts. 



In this way only (discarding as but the transient outward form of a limited 

 portion of an infinite system of ideas, all notion of extension as essential to 

 the conception of geometry, however useful as a suggestive element) we 

 may hojie to see accomplished an organic and vital development of the 

 science. 



