Mr. J. J. Sylvester on a new Class of Theorems, 363 



course. Some later observations, received from Mr. Curtis 

 at Longford, and a consideration of the effects of perspective 

 at Perth and Edinburgh, incline me to admit that the path 

 might make an angle 3° or 4° greater with the meridian than 

 I have above supposed. These conclusions are independent 

 of the actual distance or parallax of the meteor; which, as I 

 have said, cannot be determined without further observations, 

 which I should be glad to receive from any quarter, but more 

 particularly from Ireland, and from the centre and N.W. of 

 Scotland. If correct, they entitle us to infer that the meteor 

 in question was most probably a body moving in space, in a 

 path little curved, and not revolving round the earth. 



XL VI I. Additions to the articles in the September Number of 

 this Journal, " On a new Class of Theorems " and on Pascal's 

 Theorem. By J. J. Sylvester, M.A., F.R.S* 



THIRST addition. — I have alluded in the above article to a 



more general theorem, comprising, as a particular case, 



the theorem there given for the simultaneous evanescence of 



two quadratic functions of 2n letters, or n linear equations 



becoming instituted between the letters. 



In order to make this generalization intelligible, I must 

 premise a few words on the Theory of Orders, a term which 

 I have invented with particular reference to quadratic func- 

 tions, although obviously admitting of a more extended appli- 

 cation. A linear function of all the letters entering into a 

 function or system of functions under consideration I call an 

 order of the letters, or simply an Order. Now it is clear that 

 we may always consider a function of any number of letters 

 as a function of as many orders as there are letters ; but in 

 certain cases a function may be expressed in terms of a fewer 

 number of orders than it has letters, as when the general 

 characteristic function of a conic becomes that of a pair of 

 crossing lines or a pair of coincident lines, in which event it 

 loses respectively one and two orders, and so for the charac- 

 teristic of a conoid becoming that of a cone, a pair of planes 

 or two coincide! it planes, in which several events, a function 

 of four letters, becomes that of only three orders, or two 

 orders, or one order, respectively. When a function may be 

 expressed by means of r orders less than it contains letters, I 

 call it a function minus r orders. I now proceed to state my 

 theorem. 



Let U and V be functions each of the same m letters, and 



* Communicated by the Author. 



