

138 Mr. J.J. Sylvester's Enumeration of the 



Uuilineo-indefinite con- \ r^ + y^ + xz+ ijt 

 tact. Five conditions. J ^^yS^^^a^P^^^cy^ 



3rd Class. 

 Curviliueo-indetinite "1 ™2 i ,,2 i _2 i /< 

 contact. '^ 



Five conditions. 



Bilineo-indelinite "1 ,^2 ^ ^,„ ^ ^^f-j 

 contact. y " Y 



Six conditions. J xij + ztJ 



Another (and, in a physical sense, more) natural mode of 

 gronping the twelve species of conoidal contact, which, without 

 obsemng the same lines of demarcation, leaves intact the se- 

 quence of the species, is into the three families. The first, or 

 definite-continuous, for which the surfaces touch in a single 

 point, and intersect in an unbroken curve, comprises simple and 

 cuspidal contact. 



The second definite-discontinuous, for which the surfaces 

 touch in one, two, three or foiir points, but intersect in a cuiTC 

 more or less broken up into distinct pai'ts, comprises all the 

 species from the third to the ninth inclusive. The third natural 

 family is that of indefinite contact, and comprises the three last 

 species. It will of course be observed that there are five species 

 of single contact, i. e. contact at one point, viz. simple, cuspidal, 

 and the three confluent species, two of double, one of treble, one 

 of quadi'uple, and three of indefinite contact ; the last being 

 distinguishable inter se — lineo-indefinite as being special at two 

 points, curvilineo-indefinite as having no speciality, and bilineo- 

 indefinite as being special at one point only. 



I might now proceed to discuss more particularly the nature 

 of the contact taken, not collectively, but with reference to each 

 single point where it exists. This, however, must be resei-ved 

 for a futui-e communication ; as also, among other important 

 and curious matter, the a"scertainment of the singular forms of 

 quadratic conjugate functions of five or more letters. At pi-esent 

 I shall content myself with stating the following general propo- 

 sition, which naturally suggests itself from a consideration of the 

 cases already considered. 



In a conjugate quadratic system of any number of letters, the 



lowest and also the highest degree of singularity will be always 



unique ; the conditions to be satisfied in the former case being 



r — 1 

 only one in number, and in the latter r ^ , where r denotes 



the number of the letters. The first part of this proposition is 

 self-apparent, the latter part may be inferred from the homaloidal 

 law; for the (r— 2)nd minors will be quadratic functions, and 

 the highest degree of contact will correspond to those having 



