Mr. J. J. Sylvester on Equivalent Quadratic Functions. 295 



pressed in the foregoing pages has been heretofore estabUshed — 

 that of Lenz and Jacobi ; and this, as we have seen, forms a hnk 

 in a chain of laws, or rather a deduction, which flows u priori 

 from the combination of the 3rd and 4th propositions. The 

 laws of magnetic action, at distances in comparison with which 

 the thickness of the magnet vanishes, have been long known. 

 But the complementary portion of the subject, which embraces 

 the laws of action at short distances where the thickness of the 

 magnet comes fully into play, has, so far as I am aware, 

 hitherto eluded the gi-asp of experiment and formed a subject of 

 mere puzzling conjectiu-e. The want here experienced it has 

 been the object of the present inquiry to supply. 



The principal results may be summed up as follows : — 



1. The mutual attraction of a magnet and a sphere of soft 

 iron, when both are in contact, is directly proportional to the 

 strength of the magnet. 



2. The mutual attraction of a Qiagnet and a sphere of soft 

 iron, when both are separated by a small fixed distance, is directly 

 proportional to the square of the strength of the magnet. 



3. The mutual attraction of a magnet of constant strength and 

 a sphere of soft iron is inversely proportional to the distance be- 

 tween the magnet and the sphere. 



4. When the distance between the magnet and the sphere 

 varies, and a constant force opposed to the pull of the magnet is 

 applied to the latter; to hold this force in equilibrium, the 

 strength of the magnet must vary as the square root of the 

 distance. 



I have, in conclusion, to expi'ess my deep sense of the kind- 

 ness of Professor Knoblauch, who, during this investigation, 

 permitted me to occupy three of his rooms, and placed his ex- 

 tensive and beautiful collection of apparatus entu'ely at my 

 disposal. 



Marburg, January 1851. U1 . ,;ft (ir.v 



XXXVII. On the Relation between the Minor Determinants of Li- 

 nearly Equivalent Quadratic Functions. By J. J. Sylvester, 

 M.A., F.R.S.* 



I SHOWED in the preliminary part of my paper on Contacts 

 in the February Number of this Magazine, by a priori rea- 

 soning, that if a quadratic function (U) be linearly converted 

 into anf)ther (V), any minor determinant of any order of V must 

 be a syzygetic function of all the minor detei'minants of U of 

 the same (n'der. 



The object of my present communication is to exhibit the 



* CoinmuiiicutcJ by the Autlior. ,j 



