368 Mr. J. J. Sylvester on a new Class of Theorems. 



equaled to zero, constitutes the equation to such figure. 

 Plucker, I think, somewhere calls it the line or surface func- 

 tion, as the case may be. Geometry, analytically considered, 

 resolves itself into a system of rules for the construction and 

 interpretation of characteristics. One more remark, and I 

 have done. A very comprehensive theorem has been given 

 at the commencement of this commentary, for interpreting 

 the effect of a complete determinant of a linear function of two 

 quadratic functions (U + //V), having one or more pairs of 

 equal factors (e + e/ji). But here a far wider theory presents 

 itself, of which the aim should be to determine the effect and 

 meaning of this determinant, having any amount and distri- 

 bution of multiplicity whatsoever among its roots. Nor must 

 our investigations end at that point; but we must be able to 

 determine the meaning and effect of common factors, one or 

 more entering into the successive systems of minor determi- 

 nants derived from the complete determinant of U + //V. 



Nor are we necessarily confined to two, but may take several 

 quadratic functions simultaneously into account. 



Aspiring to these wide generalizations, the analysis of qua- 

 dratic functions soars to a pitch from whence it may look 

 proudly down on the feeble and vain attempts of geometry 

 proper to rise to its level or to emulate it in its flights. 



26 Lincoln's-Inn-Fields, 

 September 3, ] 850. 



The law which I have stated for assigning the number of 

 independent, or to speak more accurately, non-coevanescent 

 determinants belonging to a given system of minors, I call 

 the Homaloidal law, because it is a corollary to a proposition 

 which represents analytically the indefinite extension of a pro- 

 perty common to lines and surfaces to all loci (whether in 

 ordinary or transcendental space) of the first order, all of 

 which loci may, by an abstraction derived from the idea of 

 levelness common to straight lines and planes, be called Ho- 

 maloids. The property in question is, that neither two 

 straight lines nor two planes can have a common segment ; in 



tion which, equaled to zero, expresses the condition of the actuality of such 

 fact or existence. 



Perhaps the most important pervading principle of modern analysis, but 

 which has never hitherto been articulately expressed, is that, according to 

 which we infer, that when one fact of whatever kind is implied in another, 

 the characteristic of the first must contain as a factor the characteristic of 

 the second ; and that when two facts are mutually involved, their charac- 

 teristics will be powers of the same integral function. 



The doctrine of characteristics, applied to dependent systems of facts, 

 admits of a wide development, logical and analytical. 



