Mr. R. H. M. Bosanqiiet on the Theory of Sound. 25 



The characteristic form /(.r ^ ?/, z) may be expressed by an 

 equation of the type 



the ratio of oji and (02 being irrational, but the coefficients of 

 the quadratic forms /i and/2 being integral numbers. There 

 is a simple symmetry when the discriminantal cubic of /i + 6f2 

 has one rational root, an ellipsoidal symmetry when it has 

 three rational and unequal roots, a spheroidal symmetry when 

 it has two equal roots. (It cannot have its three roots equal, 

 because the cone /(ci', y, ^) = is imaginary.) 

 ' We suppress the further discussion of these conditions, 

 only observing that they may be so expressed as to show that 

 they depend only on the coefficients of the four given relations, 

 and not on the six coefficients A, B, C, A^, B^, C^ themselves. 



12. Case of five linear relations between the coefficients. 



In this case the ratios of the coefficients are themselves evi- 

 dently rational, and the paralielepipedal system has a spherical 

 symmetry. It is also true, conversely, that when there is a 

 spherical symmetry the ratios of the coefficients are rational. 



We may mention that the question of the rationality or 

 irrationality of the ratios of the crystallographic coefficients 

 had attracted the attention of Gauss, who, as appears from the 

 memoir of his life (Grauss, Zum Geclachtniss, von W. Sarto- 

 rius V. Waltershausen : Leipzig, 1856), had in the year 1831 

 devoted himself with great ardour to the study of crystallo- 

 graphy*. 



lY. jSfotes on the Theory of Sound. By B. H. M. BoSAN- 

 QUET, Fellow of St. Johns College, Oxford. 

 [Continued from vol, iii. p. 424] 

 5. On the Symmetrical Spherical Divergence of Sound in Air, 



THE principal interest of this subject, from a practical point 

 of view, is its bearing upon the reflexion of sound at its 

 divergence from the end of a tube. I hope to return to a 

 more particular examination of this question on a subsequent 

 occasion ; in the present note I shall allude to it only so far as 

 is necessary to make intelligible the interest of the results ob- 

 tained in the restricted case here dealt with. 



The investigations of Helmholtz on the divergence of sound 

 from the open end of a cylindrical tube {Crelle, 1860), broke 

 ground for the first time in the knowledge of the manner in 



* Some of tlie demonstrations, whicli have been omitted in the present 

 note, will be found in a paper inserted in the ' Proceedings of the London 

 Mathematical Society,' vol. vii. p. 83. 



