220 



Dr. T. Muir on Lagrange 9 s 



the maximum value for 6=co . For infinitely increasing 

 temperature <f> K would equal C/A, 5 , and the maximum of energy 

 would approach infinitely near to the wave-length zero. 



While I had deduced the formula for <£ A from the theore- 

 tical considerations just brought forward, Prof. Paschen found 

 independently that the formula 



*> 



C _£ 



where a is a constant, was the one which reproduced best the 

 results of his observations, and was kind enough to communi- 

 cate this to me and to allow me to publish his formula here. 

 Prof. Paschen intends to determine the value of the constant 

 a from a complete calculation and comparison of his experi- 

 ments. If a. is not equal to 5, the total emission would not 

 follow Stefan's law. 

 Charlottenburgh, June 1896. 



XXXI. On Lagrange's Deter minantal Equation. 

 By Thomas Muir, LL.D* 



1. ^VTAMOUS proofs t have been given of the reality of 

 T the roots of the equation 



a — x 

 b 



c 



b G . 



d — x e 



e f—x . 



0, 



and more than one extension J of the theorem has been made. 

 Apparently, however, no departure from axi- symmetry of the 

 determinant has ever been contemplated until quite recently. 

 This new and important step is due to Professor Tait, who in 

 a paper read before the Royal Society of Edinburgh in May 

 is led to the conclusion that the cubic equation 



—w 



C 



b_ 



P 



a 



9 



— x 



* Communicated by the Author. 



t For three of them see Salmon's ' Modern Higher Algebra,' 4th edit 

 pp. 28, 48-56. 



% See Sylvester, Cr die's Journal, lxxxviii. pp. 6-9. Kouth, e Dynamics 

 of a System of Kigid Bodies/ part ii. 4th edit. pp. 36-88, 41. Muir, 

 ' Messenger of Math.' xiv. pp. 141-143. 



