TRANSACTIONS OF THE SECTIONS. 5 



As another example, let a= 11, so that the two equations to be solved are 

 ar—y'=^U=z' and x^ — lly^=n = w^; then taking ;i = 5, we see that one 

 obvious solution of the two auxiliary equations a;^— y^ = 52:^and x'-~\\y'^=bur 

 is x=1 , y=2, z-=Z, and w=l ; and .". by the foregoing theorem 



X=.r2^+yV = 2P + 2^=445 and \'—2xyzw=S,^, 

 which are the least integral values of x and y to fulfil the proposed equations ; 

 they give z=437 and ty=347 ; and now from this solution we find another, 

 as indicated above, viz. new X=x*— ay*— 445*— II . 84"* and 



new Y=2xyzw=2 X 445 X 84 X 437 X 347 = &c. ; 

 and by using these values of X and Y for x and y, we can thence again 

 find X and Y in very great integers, &c. By taking a negative, we could 

 obviously deduce the solution of X^ — Y^=Z^ and X- + a6Y- = W^ from a 

 solution of the two auxiliary equations x^+ay^^xz' and bx-—y" — ?iw^. 

 Finally, we may observe that the two equations x"—y-=^D and x- — Ay'^^d 

 will be simultaneously possible whenever A is =9 — '2d-, or 50 — 2a^ or 

 49_3a2, or 81— 5a'-, or 25 — 6a^ or 64 — 7a', or 100— Ua-, or any of the 

 following integers, viz. 7, 11. 18, 19, 22, 32, 36, 37, 42, 46, 48, 56, 57, 

 61, &c. 



General Theorem. — The solution of X- + Y" = D and X'-+ fa+ l)Y- = a can 

 be obtained from a solution of the two auxiliary equations x' + y'=^nz^ and 

 «y^ + a;^ = nw'-; in fact X=a;V— yV and Y-=2xyzw will answer, as is easily 

 demonstrated. 



Another GeneralTheorem. — ThesolutionofX^— Y^=nandX^ — («+ l)Y"=a 

 can also be obtained from a solution of the two auxiliary equations 



{o?—y'^^=nz^ I 

 aa'^+y^=W2«^ J 

 or from a solution of the pair 



fy^+ ar=nz' \ 

 ly^ — ax^=nw^ J 



for in fact X=,r^tiJ-+yV and Y=2jryrw will answer, as is also easily de- 

 monstrated. 



The author states, that it is the demonstrations of the impossible cases 

 that have led to the discovery of all the foregoing general theorems for 

 solving the possible cases ; and although these demonstrations of the impos- 

 sible cases are by far the most interesting and valuable part of this Tract, 

 they are necessarily, on account of their length, omitted in the present 

 abstract ; but the Tract quite entire will be soon published. 



On a more general Theory of Analytical Geometry, including the Cartesian 

 as a particular case. By Alexander J. Ellis, B.A., F.C.P.S. 



Assuming a Roman i as the symbol for rotating through 90°, it is shown 

 that {x + \){x—i)=^x' + \, and that therefore i= V'(— 1). Taking (joH-ij)L 

 as the representative of a line of the length a^C^^ + j^ . L, L being the unit 

 of length, making an angle with the axis, where v p" + ?"-sin d=^q and 

 V jj^-l- j-.cos0=j>, this line will determine its extreme point when referred 

 to a known origin, axis, and scale. 



Two Dimensions. 



Simple Locus. — If g be a function of ^, then the two expressions /? -|- ij and 

 q=gp determine a curve. This corresponds with the usual Cartesian case. 



