43(3 Dr. Macfarlane'a Analysis of Relationships. 



fourth combination-tone, are recorded by Konig in the neigh- 

 bourhood of the interval c : d" 



54. Beats of the mistimed consonance 1 : 7 are recorded 

 by Konig. These might be produced by a sixth combina- 

 tion-tone (difference-tone of form Gp—q) of the primaries 

 n and 7n±m. 



55. Beats of the mistuned consonance 1 : 8 are recorded. 

 These might be produced by a seventh combination-tone 

 (difference-tone of form 7p—q) of the primaries Sn±m. 



56. As far as my own experience goes, however, I have no 

 direct and palpable evidence of beats of mistuned consonances 

 higher than 1 : 4, or of the existence of combination-tones 

 higher than the third (3p—q) in recognizable intensity. Up 

 to this point the phenomena are quite clear ; and there is no 

 possible doubt as to their nature. 



But in considering these limited results it must be remem- 

 bered, (1) that I have restricted myself to notes of very mode- 

 rate intensity, so that the phenomena might correspond as 

 nearly as possible to those which are presented to our ears in 

 practice, and (2) that, although I was unable to get rid en- 

 tirely of the presence of upper partials in all cases, yet the 

 phenomena were subjected to a careful and prolonged analysis 

 by listening under varied conditions, until the effect of the 

 upper partials could be separated out and eliminated with cer- 

 tainty. And we have at all events no security that these upper 

 partials did not give rise to many of Konig's results; indeed it 

 is almost certain that they must have entered into those results. 



Note. — The present paper was written before the appearance 

 of Konig's paper in Wiedemann's Annalen in the present 

 year. The discussion of that paper, though necessary for a 

 complete view of the subject, must be reserved till after the 

 conclusion of the present paper. 



[To be continued.] 



LVIIL An Analysis of Relationships. 

 By A. Macfarlane, M.A., D.Sc, F.R.S.E* 



IN this article I propose to describe some results of several 

 papers on an Algebra of Relationship, which I have 

 recently contributed to the Royal Society of Edinburghf. 

 The Logic of Relatives J has been worked at by De Morgan, 

 Leslie Ellis, Harley, and C. S. Peirce§ ; and the last-named 



* Communicated by the Author. 



t Proc. Roy. Soc. Edinb. May 1879, Dec. 1880, and March 1881. 



X Since writing this article I have had the opportunity of reading two 

 interesting and suggestive papers on the Logic of Relatives, by Mr. J. J. 

 Murphy. 



§ For references see Jevons's i Principles of Science/ p, 23, 



