Theorem relating to Matrices. 173 



observed ; and M. H. Becquerel (Annates de Chimie et cle 

 Physique, 1883, tome 30) gives the wave-length of the 

 longest band known to him as 1 M, 5. These remarks will not 

 be superfluous as an introduction to the preceding table, 

 which presents a summary view of the advances made beyond 

 the above-named point in the last five years. 



Broadly speaking, we have learned, through the present 

 measures with certainty, of wave-lengths greater than 0*005 

 millim., and have grounds for estimating that we have recog- 

 nized radiations whose wave-length exceeds 0*03 millim. ; so 

 that while we have directly measured to nearly 8 times the 

 wave-length known to Newton, we have probable indications 

 of wave-lengths far greater, and the gulf between the shortest 

 vibration of sound and the longest known vibration of the 

 ether is now in some measure bridged over. 



In closing this memoir I would add that the very consider- 

 able special expenses which have been needed to carry on 

 such a research, have been met by the generosity of a citizen 

 of Pittsburgh, who in this case, as in others, has been content 

 to promote a useful end, without desiring publicity for his 

 name. 



I cannot too gratefully acknowledge my constant obliga- 

 tion to the aid of Mr. F. W. Very and Mr. J. E. Keeler of 

 this Observatory, who have laboured with me throughout this 

 long work. In the prolonged numerical and other com- 

 putations rendered necessary, I have been aided by Prof. 

 Hodgkins of Washington, and by Mr. James Page of this 

 Observatory. 



Allegheny Observatory, 

 May 31, 1886. 



XX. An Extension of a Theorem of Professor Sylvester's 

 relating to Matrices. By A. Buchheim, M.A* 



ONE of Prof. Sylvester's fundamental theorems in the 

 theory of matrices is what he calls the interpolation- 

 formula : viz. if m be a matrix of order n, and X : . . . X„ its 

 latent roots, we have, c/> being any function, 



v (m — X 2 )(m — X 3 ) . . . (m — \ n ) 



This theorem only applies so long as the latent roots are un- 

 equal. In this note I extend it to matrices having equalities 



* Communicated by the Author. 



