TRANSACTIONS OF THE SECTIONS. 



3 



When A=7, then taking n=2, one obvious solution of the auxiliary eqaa- 

 tions a'" + 7y'=2z^ and x- — 7f=2w'^ is a;=5, y = l, z=4, and w = 3; and 

 hence by the above general theorem, we find X = in(z'^ + w*) = 4^ + 3* = 337 

 and Y=2xyzw=l'20 to fulfil the two proposed equations ar+ 7y-=U=2F and 

 x^—7y'-=n = w-, giving z — 463 and w = \l3; and thence vi^e find again, 

 according to the above observation, new a;=|^(463''+113^) and 

 new; y = 337 X 240 X 463 X 113, 



from -which we could again find values of x and y in integers still larger, &c. 

 When A=13, then taking «=1, one obvious solution of the auxiliary 

 equations x"+l3y'=z' and x-—l3y^=—w" is x=6, y = 5, giving z=l9 and 

 u;=17 ; and hence by the general theorem, we find X=|^(iy*+ 17^) = 106921 

 and ¥=10x6x19x17 = 19380 to fulfil the two proposed equations, 

 a;'-+13y^=D = z^ and a;-— 13y-=n = M'^. These values of x and y give 

 r= 127729 and «;= 80929, from which again we find, according to the fore- 

 going observation, new a'=-J(127729^+80929-') and 



newy=2 x 106921 x 19380 X 127729 x 80929, &c. 



Finally, it is observed that the solution of X" + a6 Y" = D = Z" and 

 X" — a6Y'-=n = W- can be also derived from a solution of the auxiliary 

 equations x^ + y^= az' and x-—y^=by)^, since in fact X^=x* + y* and 

 Y^2a:'yzw will answer ; for then 



aJY"=4a6^^y-2V" = 4x^y-(az-) {bw") =.4ary-{x^—y^) = 2tv 

 where t=^x*—y^ and v = 2x"y^, and X^=(x'* + y*)'=^f + v-; and so 

 X^±abY^={t±vf, which are both squares. Q. E. D. 



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

 taneous equations a-" +y^=n anda?- + Ay"=n. Now in the original paper it is 

 rigorously demonstrated by one uniform, easy, and satisfactorj^ method, that 

 thi.sis impossible when A is any positive integer < 20, except 7, 10, 11 or 17; 

 and it is also satisfactorily proved that the proposed equations will be always 

 possible or solvable whenever A is =2a^— 8, or 2a^— 1, or 2a^-f-2, or 2a^ + 9, 

 or 2a^+50, or 3a^— 48, or 3a^— 3, or 3«^+4, or 3a^+49, or 5a^— 4, or 



5a^+5, or 5a"— 80, or 5a==4-81, or 6a^— 2, or 6aH 3, or 4a=+ 3a, or--, 



diminished either by ^ or by 1^, &c. &c. And thus the proposed equations will 

 be possible or soluble whenever A is any of the following integers ; viz. 7, 10, 

 11. 17, 20, 22, 24, 27, 30, 31, 34, 41, 42, 45, 49, 50, 52, 57, 58, 59, 60, 

 61, 68, 71, 72, 74, 76, 79, 82, 85, 86, 90, 92, 94, 97, 99, 100, 101, 104, 

 105, 112, 115, 119. 120, 121, 122, &c. 



The solutions of the possible cases are inferred with great facility in the 

 present paper from the following new and useful — 



General Theorem. — The values of X and Y in X^-fY^=n=Z^ and 

 X^-|-a6Y^=n = W^ can be deduced or inferred from the values of x and y in 

 the auxiliary equations x" + ay^=:nz^ and y' + bx^=:nw^ ; in fact, X=x^w' — yV 

 anAY=2xyzw will answer; for then X- + Y^=(^w^+yV)^. And so the 

 first condition is fulfilled. NownX=:x'(y^ + bx^) — y^(x^ + ay-), .'. =bx* — ay*; 

 also n-Y'- = 4x-y'"-{x" + ay')(y' + bx'), .'. =4x*y*(l+ab)+4bx^y^ + 4ax-y^ ; and 

 so n-(X'- + abY") = (bx* + 'labx'^y'^ -\- ay^)~ ; and hence 



X- + abY-=(J}x*Jr2abxY + ayy-^n^' •'■ =n. 



And thus these values of X and Y satisfy the second condition also. Q. E. D. 



If a or 6 be negative, we obtain a solution of X^ + Y-:=n and X^ — a6Y"=D ; 



but by taking 6 = 1 and «= 1 , and interchanging z and w, this general theorem 



shows that, "from one solution of the proposed equations a;" +y^= 2^ and 



1* 



