2 REPORT — 1855. 



known theorem, which I call the porism (allographic) of the in-and-circumscribed 

 triangle, viz. " there are an infinity of triangles inscribed in a conic, and such that the 

 sides touch conies, each of them meeting the circumscribed conic in the same four 

 points." 



The investigations, the results of which have just been stated, will appear in the 

 Quarterly Mathematical Journal. 



A Tract on the possible and impossible cases of Quadratic Duplicate Equa- 

 lities in the Diophantine Analysis. By Matthew Collins, B.A., Senior 

 Moderator in Mathematics and Physics, and Bishop Latvs Mathematical 

 Prizeman, Trinity College, Dublin. 



The author of this tract divides it into three chapters. 



Chapter I. treats of the possible and impossible cases of the two simul- 

 taneous equations x--\-ky-=-ll and x' — Ay"=D ; now it is proved in the 

 original paper from which the present abstract is taken that this is impossible 

 Avben A is any integer < 20, except 5, 6, 7, 13, 14 or 15. And the demon- 

 strations of the impossibility are extremely easy, clear, and rigorous, and pos- 

 sess the great advantage of being effected, in all the different cases, by one 

 uniform method. This first chapter terminates with a ^renera/ demonstration 

 of the impossibility whenever A is a prime number, and such that neither 

 m~-\-\ nor m' — 2 is divisible by A, m being ^^^A. 



In the cases that are possible, as many solutions as we please, in integers 

 {x, y) prime to each other, are obtained in this paper with singular facility 

 and rapidity by means of the following new and useful — 



GraeraZTZ/eorem. —ThesolutionofXHa6Y== D =Z=andX=— a6Y-= n =W= 

 can be obtained from a solution of the two auxiliary equations ax' + by'=nz- 

 and abx'—y-=+nw'^, for in fact X = |«(2^-f-2<;^) and Y=^2xyzw will answer. 



Demonstration. — The difference of the squares of the two auxiliary equa- 

 tions gives 4oJ.r-y"=K-(2^—w^), and .•. abY'-=^4abx"y-z-w-, .'. =n-z'-w-{z*—iv^); 

 and as 4X-=n-{z'^ + w^y=^n-(z'^—w'^y- + n\2z-zo-y- = n-{t'--\-v-), where t = z*—w* 

 andv = 2^ur and 4abY'- is =4n'z-uf{z*—w*), .•. =n-{2tv), 



.'. 4(X" + abY')^n-(t + v)", which are both squares. Q. E. D. 



By taking ?«=1 and also i=l, we can, from one solution of the equations 

 3r + ay"=z- and :r — ay--=-vr, derive another solution of the same equations in 

 larger integers ; thus new X=:^(2^-fi<;'') and new Y=-2xyzw. 



Ex.gr. When A=5, then the auxiliary equations ,r--f5y"=«z" and 

 x^ — 5y"= — nur are obviously fulfilled by taking w=1=3/=m>, a: = 2 and z-=Z; 

 hence by the general theorem, we find X = |^(s^ -|- «<;'') = |^(3^-f- 1^) = 41 and 

 Y = 2a'y2J<'=12 to fulfil the proposed equations 



■r"-f-5y"= n =2" and xr — 5y"= n =w", 



giving 2=49 and to-=Z\ ; and from this set of answers we can, according to 

 the above observation, deduce another set in larger integers ; in fact, it is 

 evident new 



^=i(49'4-31^) = 3344161, and new y=2 x 41 x 12 x 49 x 31 = 1494696, 

 from which we could again find new and very high values of x andy, end thus 

 ascend into very great whole numbers. 



When A=:6, then ^=5 and y = 2 give 2= 7 and w=l; 



.•.newx=iCl^ + V) = UO\, 



and new y=10 x 2 x 7 = 140, giving new 2=1249 and new m;=1151, and 

 thence again 



A'^r'w^=i(1249'-M151-') andneM;y=1201 x 2S0x 1249 x 1151, &c. 



