CHAP, xix] SYSTEMS OF EQUATIONS OF DEGREE Two. 527 



Then cx = u 4 +2u-z-z 2 = (mz-u 1 } 2 if z = 2w 2 (ra+l)/(ra 2 +l). S. Jones took 

 /3= (2z y}- and found 2(n 2 ari)k, 2(a+ii)k, 2k 2 , where k = a' 2 -\-n-. 



W. Wallace 1400 made a=xy+z*, $=xz+y\ 7 = yz+x 2 , and , 1 / 2 + j 8 1/2 +7 1/2 

 squares by taking a = (2y+ z) 2 , (3 = (2z+y) 2 , whence x = 4 (y+z). Then y = r- 

 if ?/2 = II{r4(?/+s)}. Equate the factors to ymfn and zrc/m. We get y, z 

 and hence x as rational functions of m, n, r. Omitting the common 

 denominator, we have :r = 4(??i 2 +n 2 )r, y=(Smn+n z )r, z = (m 2 8mn)r. 

 Then a, 0, 7 equal the squares of (??i 2 +8mn+2n 2 )r, (2?w 2 8wm+w 2 )r, 

 (4?/z 2 +?nn 4n 2 )r. The sum (7m?+mn n 2 )rof these is a square if r equals 

 the first factor or the quotient of it by any square. 



MISCELLANEOUS SYSTEMS OF EQUATIONS OF DEGREE Two. 



Diophantus, III, 2, made s 2 -\-Xi (i = l, 2, 3) rational squares, where 

 s = Xi+x 2 -{-X3. In Diophantus, III, 3, s 2 #,- (i = l, 2, 3) are made squares. 

 T. Brancker 141 treated the latter problem. A. Gerardin 142 gave several 

 integral solutions of the last two problems. 



Diophantus, III, 4, made a;,- s 2 (i=l, 2, 3) rational squares. 



To find xi Xz - - such that 



where s = 2Xi, "Comes" 143 noted that since p 2 i} s 2 , q* are squares in arith- 

 metical progression we may use the known values 



p t - = s(m] n* +2171^1(171* +rii), 3. = s(nlm 2 i +2m i n l : )/(m 2 i +nl') . 



Then s = Zz; gives s. For Diophantus' solution, see the first page of Ch. VI. 



A. Gerardin and R. Goormaghtigh 144 made s 2 x'\ (i=l, 2, 3) squares; 



also s 2 (s x^; also s 2 (sxi) (i=I, 2, 3, 4), where s = Xi+ - -+4. 



The latter 145 " made s 2 +x,- (i = l, , n) squares, also s 2 (s a?,-), where 



S = XiH ----- \-X n - 



Leonardo Pisano 146 treated cases of x\-\rXi-\ ----- \-x n =yi, yl+xl yl, 



yl+xl=yl, ,2/-i+z=2& 



J. Cunliffe 147 made a-\-Xi (i = l, 2, 3) squares, where cr = xl+xl-\-xl. 



S. Ryley 148 made a = x z +yz-\-y 2 , P=x 2 +yz+z 2 , y = y 2 +yz+z 2 squares. 

 Take a = a 2 , (3 = b 2 . Then y 2 -z 2 = a?-W. Hence take (a+b)r=(y+z)s, 

 (afys = (yz)r, which give a, b in terms of y, z. Now 7= D if 



Then aPyz y 12 becomes a function of r, s, m, n of degree 4 in w, which will 



140a New Series of Math. Repository (ed., T. Leybourn), 3, 1814, I, 21-23. 



141 An Introduction to Algebra, transl. out of the High-Dutch by T. Brancker, much altered 



and augmented by D. P[ell], London, 1668, 102-4. 



142 L'intermediaire des math., 22, 1915, 197-8. 



143 The Gentleman's Math. Companion, London, 4, No. 21, 1818, 752-7. 



144 L'intermediaire des math., 22, 1915, 220-1, 244; 23, 1916, 136-141, 155-7, 209-11; 24, 



1917, 13-14. 



146 Nouv. Ann. Math., (4), 16, 1916, 401-26. 



146 Scritti di L. Pisano, 2, 1862, 279-83. Cf. F. Woepcke, Jour, de Math., 20, 1855, 61-62; 

 A. Genocchi, Annali Sc. Mat. Fis., 6, 1855, 193-205, 357-9. 



147 Math. Repository (ed., Leybourn), London, 3, 1804, 97-106. 



148 The Gentleman's Math. Companion, London, 1, No. 8, 1805, 42-4. 



