from Equations of Coexistence. 433 



Cor. Hence, in place of Sturm's auxiliary functions, we 

 may substitute the functions derived from the equations of 



/ dfx \ 



coexistence \fx = 0, -y— = 01 according to theorem (2.) 



due regard being had to the sign. 



Scholium. Hitherto it has been supposed that the values of 

 the coefficients in the equations of coexistence are independ- 

 ent of one another, but particular relations may be supposed 

 to exist which shall cause the leading terms given by theorem 

 (2.) to vanish, giving rise to anormal or singular primes, as 

 they may be called, of the degree /' of fewer than {m — r). 

 (71 — r) dimensions. The theory of this, the failing case, (so 

 to say) is highly interesting, and I have already discovered 

 the law of formation for the quotients of succession on the 

 supposition of «;«/ nuinher of primes vanishing consecutively; 

 but I forbear to vex the patience of my reader further, the 

 more so, as I hope soon to be able to present a complete me- 

 moir, with all the steps here indicated filled up, and nume- 

 rous important additions, (the perfect image of which this is 

 but a rough mould,) as homage to the learned and illus- 

 trious society which has lately done me the honour of ad- 

 mitting me into its ranks. 



Why this has not already been done must be excused, by 

 the fact of the theory having suggested itself abroad in the in- 

 tervals ofsickness*. Yet thus much will I add in general terms, 

 viz, that as many primes as vanish consecutively, so many units 

 must be added to the index 2 of the accessions received in 

 the numerator and denominator of the subsequent quotient; 

 and in the quotient after that, it is not the square of the lead- 

 ing term of the penultimate prime, — but the product of this 



ing possible, a litde consideration will serve to show that the leading term 

 of each prime derivative of the = w Ifx -r- f = will consist of a se- 

 ries of fractions, each of which fractions is, numencally speaking, of the 

 $ame sign. 



* The reflections which Sturm's memorable theorem had originally ex- 

 cited, were revived by happening to be present at a sitting of the French 

 Institute, where a letter was read from the Minister of Public Instruction, 

 requesting an opinion upon the expediency of forming tables of elimination 

 between two equations as high as the 5th or 6th degree containing one 

 repeating term. 'J'he offer was rejected, on the ground of the excessive 

 labour that would be required. I think that this has been very much over- 

 rated ; and probably many will be of the same opinion who have dwelt upon 

 the fact that no numerical quantity will occur in the result higher than 

 the highest index of the repeating term. Would it not redound to the 

 honour of British science that some painstaking ingenious person" should 

 gird himself to the task? and would not this be a proper object to meet 

 with encouragement from the Scientific Association of Great Britain? 



Phil. Mag. S. 3. Vol. 15. No. 98. Dec. 1839. 2 F 



