20 Mr. W. Stevenson on the Aurora, and the connexion which 



fx and <f>x vanish ; in which case, instead of the coefficients being 

 linear, some of them will be, as in fact all might be, polynomial 

 functions of x. The rule, it may be proved, will still subsist. 



Equating the first and last quotients (each of them to 4 1 and 

 to —1, and the intermediate ones to 4-2 and to —2), the 

 greatest root of all the equations so formed continues to be a 

 superior, and the least root an inferior limit to the roots of fx. 

 Nor is it ever necessary, even in these special cases, actually to 

 solve any of these equations ; for evidently it will be sufficient to 

 find a superior limit and an inferior limit to each of thern, and 

 adopt the greatest of the superior and the least of the inferior 

 limits as the superior and inferior limit to the roots of the given 

 equation. Thus, then, we should have to repeat upon the quo- 

 tients increased and diminished by 1 or 2 (as the case may be), 

 the same process as is supposed to be originally applied to fx, 

 and thus by a continued process of trituration (since every new 

 function so to be operated upon is of a lower degree than the 

 original function) we must finally descend to linear equations 

 exclusively. 



It is interesting thus to see that there are no failing cases in 

 the application of the rule, and that a solution of equations of a 

 higher degree than the first is never necessary. But as a matter 

 of fact, the chances are infinitely improbable (if <p{x) is chosen 

 at random), of any of the quotients after the first ceasing to be 

 linear ; and the first is of course linear, provided that the degree 

 of <j)(x) is taken only one unit below that of fx. 



In working with Sturm's theorem, a system of quotients is 

 supplied ready to hand ; and these quotients, by virtue of the 

 rule given above, may be used to assign a superior and inferior 

 limit in the first instance, before setting about to determine the 

 distribution of the roots between these limits by aid either of 

 these same quotients or of the residues. For the change of sign 

 of the residues required by the Sturmian process will only affect 

 the signs, and not the forms of the quotients ; but in the appli- 

 cation of the above rule for finding the limits, the sign of any 

 quotient is evidently immaterial. 



IV. Abstract of Observations on the Aurora, Cirri, fyc. made at 

 Dunse. By William Stevenson, Esq,* 



THE following abstract of observations and notes regarding 

 the aurora, and the connexion which appears to subsist 

 between it and the formation and modification of clouds, parti- 



* Communicated by Professor Faraday. 



