of SivLYm's Theorem. ' ' '' 45S 



unaltered in the 1st, 3rd, 5th, &c. places; whereas the residues 

 or excesses change their signs in the 1st and 2nd, 5th and 6th, 

 9th and 10th, &c., and remain unaltered in the 3rd and 4th, 

 7th and 8th, 11th and 12th, &c. places. The effect is, that if, in 

 applying Sturm's method, we omit to change the signs of the 

 remainders, and take as our signaletic series 



= i.^to4^ M fx,fx, Ri, R^, Rg, . . . R,_„ >^i«irem 



Rj, Rg, R3, &c. being the successive unaltered residues, the sig- 

 naletic index corresponding to any value of x instead of being 

 the number of continuations in the above series, will become the 

 number of continuations in going from a term in an odd place 

 to a term in an even place plus the number of variations in going 

 from a term in an odd place to a term in an even place. 



If we adopt the quotient method, the rule will be simply to 

 change the sign of the alternate quotients (beginning with the 

 second) in forming the signaletic series. 



As an artist delights in recalling the particular time and atmo- 

 spheric effects under which he has composed a favourite sketch, 

 so I hope to be excused putting upon record that it was in listen- 

 ing to one of the magnificent choruses in the ' Israel in Egypt,' 

 that unsought and unsolicited, like a ray of light, silently stole 

 into my mind the idea (simple, but previously unperceived) of 

 the equivalence of the Sturmian residues to the denominator 

 series formed by the reverse convergents. This idea was just 

 what was wanting, — the \^^rppt^y% t)i^,du^j§^d perfec^.evpliitip^ 

 of the theory. ^ ^', % ' ; """ ^' ' oiw^^m a^ 



Postscript, 



Immediately after leaving the foregoing matter in the hands 

 of the printer, a most simple and complete proof has occurred to 

 me of the theorem left undemonstrated in the text. 



Suppose that we have any series of terms w^, u^, Wg, . . . w„, 

 where 



Mi=Ai WgcsAjAg— 1 %=AiA2A3— Aj— A3 &c.^ 

 and in general n#i=r,o 



then Ui, u^y %, . . • u^ will be the successive principal coaxal deter- 

 minants of asymmetrical matrix. Thus suppose 7i=5; if we 

 write down the matrix 



aiKfiio'f him .^a-^rJii "It)' 



