24 REPORT — 1841. 



to call " Sturm's Determinators," and proceeds to lay bare the internal anatomy 

 of these remarkable forms. 



He uses the Greek letter " ^" to indicate that the squared product of the diflfer- 

 ences of the letters before which it is prefixed is to be taken. 



Let the roots of the equation be called respectively a, b, c, e . . . I, the determina- 

 tions taken in the inverse order are as follows : — 

 ^ {a, b, c, e . . . I). 



2 ^ (i, c, e . . . a; — 2 o ^ (6, c, e . . . Z). 

 2^(c, e. . .Z)a:2_2(a+6).CCC'e. . .Z)a; + 2a,6.^(c, e,...Z). 



2 (^(k,l) (x—a) (x—b) (x — c) (x—e) . .. (» — A)). 

 It may be here remarked, that the work of assigning the total number of real and 

 of imaginary roots falls exclusively upon the coefficients of the leading terms, which 

 the author proposes to call " Sturm's Superiors : " these superiors are only partial 

 symmetric functions of the squared differences, but complete symmetric functions of the 

 roots themselves, differing in the former respect from those other (at first sight similar- 

 looking) functions of the squared differences of the roots, in which, from the time of 

 Waring downwards, the conditions of reality' have been sought for. It seems to have 

 escaped observation, that the series of terms constituting any one of the coefficients 

 in the equation of the squares of the differences (with the exception of the first and 

 last) each admit of being separated and classified into various subordinate groups in 

 such a way, that instead of being treated as a single symmetric function of the roots, 

 they ought to be viewed as aggregates of many. In fact, Sturm's superior No. 1. 

 is identical with Waring's coefficient No. 1 ; Sturm's superior No. 2. is a part of 

 Waring's coefficient No. 3 ; Sturm's superior No. 3. is apart of Waring's coefficient 

 No. 6 ; arid so forth till we come to Sturm's final superior, which is again coextensive 

 and identical with the last coefficient in the equation of the squares of the differences. 

 The theory of symmetric functions of forms which are themselves symmetric func- 

 tions of simple letters, or even of other forms, the author states his belief is here 

 for the first time shadowed forth, but would be beside his present object to enter 

 further into. He concluded with calling attention to the importance to the general 

 interests of algebraical and arithmetical science that a searching investigation should 

 be instituted for showing, a priori, how, when a set of quantities is known to be 

 made up partly of possible and partly of pairs of impossible values, symmetrical 

 functions of these, one less in number than the quantities themselves, may be 

 formed, from the signs of the ratios of which to unity and to one another the 

 respective amounts of possible and impossible quantities may at once be inferred : 

 in short, we ought not to rest satisfied, until, from the very form of Sturm's Deter- 

 minators, without caring to know how they have been obtained, we are able to 

 pronounce upon the uses to which they may be applied. 



On the Theoretical Computation of Refractive Indices. By Prof. Powell. 



In the Report on Refractive Indices which the author had presented to the Asso- 

 ciation, his professed object extended only to exhibiting the results of observation, 

 without any reference to theory. That the comparison of these results with theory 

 is highly important is indeed manifest ; and the necessity for it is dwelt upon 

 (in reference to the Report just mentioned) in the address' of the General Secre- 

 taries at the meeting of 1840. It would, however (on several grounds), have been 

 foreign to the design of that Report to have introduced the subject of theoretical com- 

 putation. The author has since that period been devoting his attention to this latter 

 subject ; and it is the object of the present communication to state very briefly the 

 progress made in it. The results in the Report on Indices are classified under three 

 heads : 1st, those of Fraunhofer ; 2nd, those of Rudberg ; 3rd, those derived from the 

 latest observations of the author, comprising many tieiv results, superseding former 

 ones, and others, the combined results of several sets of earlier observations compared 

 with later. The 1st series was compared with theory, 1st, by the author in the Phil. 

 Trans. 1835, but only by an approximative and tentative method; 2nd]y, by Mr. 

 Kelland, by a direct and exact method in the Camb. Trans., vol. vi. ; Srdly, for the 

 rays D and C only, by Sir W. R. Hamilton, in the Phil. Mag., 3rd Series, vol. viii. ; 



