M. Duhamel on the Multiple Sounds of Bodies. 415 



the analysis of equations, considerable facilities will often be 

 introduced into the numerical process. I shall only observe, 

 in conclusion, that, by using other conjugate forms, it is easy 

 to see how equations of an odd degree may be brought, in a 

 direct manner, under the operation of the above method ; or, 

 without using any change of form at all, the method becomes 

 immediately applicable upon introducing, into an equation of 

 an odd degree, the new root .r = 0; or by supposing this to be 

 done after the partial analysis of the equation in its original 

 state. 



But the general principle here sketched, admits of import- 

 ant modification. I shall hereafter show that, for an equation 

 of the degree 2w, the polynomial under the radical may always 

 be reduced to the degree « — 1 ; and that for an equation of 

 the degree 2«+ 1, it may be reduced to the degree n+lx so 

 that the analysis of a biquadratic equation will depend on the 

 examination of an expression of the first degree; that of an 

 equation of the fifth or sixth degree, upon the examination of 

 an expression of the second degree ; and so on. 



In what is stated above, I have alluded to certain circum- 

 stances of doubt, of which no mention is made in the general 

 announcement in this last paragraph. I have done so because, 

 in this brief sketch of the initial steps of the investigation in 

 which I am engaged, no explicit guidance in these circum- 

 stances is actually furnished. My present purpose is simply 

 to make known the path I am pursuing in an important in- 

 quiry; and to point to the results which I confidently expect 

 to arrive at. 



Belfast, May 12, 1849. 



LXII. On the Multiple Sounds of Bodies. 

 By M. Duhamel, Member of the Institute*. 



THE question under consideration is far from being new, 

 and still it may be said to be as yet unsolved ; that is to 

 say, scientific men have not yet arrived at a common and un- 

 contested opinion on this point. The object of the present 

 memoir is to establish such an opinion. 



What follows will perhaps appear so evident, and so little 

 different from what is already known, that persons who have 

 for a long time been too easily satisfied on this subject, may, 

 whilst they accept my explanation, remain persuaded that 

 they have never viewed the matter otherwise. I expect this, 

 and consent, if they will have it so, that all my ideas and ex- 



♦ From the Annates de Chimie et de Physiqtie for January 1849. 



