Mr. J. J. Sylvester on a new Class of Theorems. 369 



other words, if n independent relations of recti! inearity or of 

 coplanarity, as the case may he, exist between triadic groups 

 of a series of 7* + 2, or between tetradic groups of a series of 

 w-f-3 points respectively, then every triad or tetrad of the 

 series, according to the respective suppositions made, will be 

 in rectilinear or in plane order. So, too, if n independent 

 relations of coincidence exist between the duads formed out of 

 n+\ points, every duad will constitute a coincidence. 



This homaloidal law has not been stated in the above com- 

 mentary in its form of greatest generality. For this purpose 

 we must commence, not with a square, but with an oblong 

 arrangement of terms consisting, suppose, of m lines and n 

 columns. This will not in itself represent a determinant, but 

 is, as it were, a Matrix out of which we may form various 

 systems of determinants by fixing upon a number p, and se- 

 lecting at will p lines and p columns, the squares corresponding 

 to which may be termed determinants of the pth. order. We 

 have, then, the following proposition, The number of unco- 

 evanescent determinants constituting a system of the pth order 

 derived from a given matrix, n terms broad and m terms deep, 

 may equal, but can never exceed the number 



(n— p-{i)(m—£ + l). 



Remark on Pascal's and Brianchon's Theorems. 



I omitte i to state, in the September Number of the Journal, 

 that the demonstration there given by me for Pascal's, ap- 

 plied equally to Brianchon's theorem. This remark is of 

 the more importance, because the fault of the analytical de- 

 monstrations hitherto given of these theorems has been, that 

 they make Brianchon's a consequence of Pascal's, instead of 

 causing the two to flow simultaneously from the application 

 of the same principles. No demonstration can be held valid 

 in method, or as touching the essence of the subject-matter, 

 in which the indifference of the duadic law is departed from. 

 Until these recent times, the analytic method of geometry, as 

 given by Descartes, had been suffered to go on halting as it 

 were on one foot. To Pliicker was reserved the honour of set- 

 ting it firmly on its two equal supports by supplying the com- 

 plementary system of coordinates. This invention, however, 

 had become inevitable, after the profound views promulgated 

 by Steiner, in the introduction to his Geometry, had once taken 

 hold of the minds of mathematicians. To make the demon- 

 stration in the article referred to apply, totidem Uteris, to 

 Brianchon's theorem (recourse being had to the correlative 

 system of coordinates), it is only needful to consider U as the 



Phil, Mag. S. 3. Vol 37. No." 251. Nov. 1850. 2 B 



