Mr. G. Shaw on some Photographic Phcenometia. 445 



and the three roots of the equation are 



2h, h — 3k, h-\-Sk respectively, 

 and are therefore all rational. 



Here it may be observed that the condition of R being an 

 even number, which we know, a priori, is the case when all 

 the roots are rational, is involved in the two more general con- 

 ditions already expressed. It will now be evident that the 

 first condition by no means involves the second, as it is per- 

 fectly easy to satisfy the equation /^ + 3g^ = Q^ without sup- 

 posing anything relative to k, the common measure of^ g, Q, 

 except that it be itself of the form X^ + Sjw.^ which will give 



an equation which can be solved in rational terms for all values 

 of A, /x, 7',s; and consequently the product of the squares of 

 the differences of the roots may be a square, and at the same 

 time the roots themselves may be irrational *. 



I believe it will be found on inquiry that the equation 

 a,**—q.v + r=zO will always have two rational roots if 



(„_l)«-i.<^«_w'*.r»-i 

 be a complete square, provided that q be prime to r. 



Furthermore, viewing the striking analogy of the general 

 nature of the conditions of rationality already obtained, to 

 those which serve to determine the reality of the roots of equa- 

 tions, I am strongly of opinion that a theorem remains to be 

 discovered, which will enable us to pronounce on the exist- 

 ence of integer, as Sturm's theorem on that of possible roots of 

 a complete equation of any degree : the analogy of the two 

 cases fails however in this respect, that while imaginary roots 

 enter an equation in pairs, irrational roots are limited to 

 entering in groups, each containing two or more. 

 4 Park Street, Grosvenor Square, 

 November 7, 1844. 



LXXV. On some Photographic Phcenomena. 

 By George Shaw, P,sq.\ 



TT is well known that the impression produced by light on 

 -*• a plate of silver, rendered sensitive by M. Daguerre's 

 process, is wholly destroyed by a momentary exposure of the 



• Thus then it appears that the total rationality of the roots of the equa- 

 tion 3^ — qx'— r = may be determined by a direct method without having 

 recourse to the method of divisors to determine the roots themselves; 

 the two conditions being that 49^ — 27r^ shall be a perfect square, and the 

 greatest common measure of ^^ and r^ a perfect sixth power. 



f Communicated by the Author. 



