CHAPTER XVIII. 



THREE OR MORE QUADRATIC FUNCTIONS OF ONE OR TWO 



UNKNOWNS MADE SQUARES. 



a 2 x'-{-dx, b'~x 2 -}-cx, MADE SQUARES. 

 J. Cunliffe 1 took v 2 -\-mv = (d v} 2 , whence v = d 2 /(2d+m). Then 



v 2 -\-nv = D 

 if d z +2dn+mn = (q d) 2 , which gives d. Then v 2 +pv = D if 



(q 2 mri) 2 +4:p(q+ri)(q 2 mri) -\-4mp (q+ri) 2 = D = (q 2 2pq mn) 2 , 



whence q = (p mn)f2. 



W. Wright 2 equated the numerators and denominators of the two values 

 of d given by d 2 +2dn+mn = (q-d} 2 and d 2 +2dp+mp = (t-d) 2 . Thus 

 q 2 mn = t 2 pm,n- s rq = p-\-t. By division, q+t= m. Hence 



q = (p m ri)/2. 

 A. B. Evans 3 made k 2 x 2 kx a square for k = a, b, c. From 



a 2 x 2 ax = (a m) 2 x 2 

 we get x. Then b 2 x 2 bx = D , c 2 x 2 ex = D if 



a 2 b 2 c 2 -abc 2 d = y 2 , a 2 b 2 c 2 -ab 2 cd = z 2 , d = 2am-m 2 . 



Subtract and set bc(2a m)=y-\-z, m(abac)=yz. Substitute the re- 

 sulting y into a 2 b 2 c 2 abc 2 d = y 2 ', we get 



bc-\-ac) 



J. Matteson 4 solved d z +2dn+mn = A 2 , d 2 +2dp+mp = B 2 by taking 

 = A-\-B, n p = AB. Inserting the resulting value of A into the 

 first of the initial equations, we get d rationally. An equal value of d is 

 obtained by use of B. It is stated that if m, n, p be any three of the numbers 

 2016, 3000, 3696, 4056 (or any three of certain 13 numbers of 6 or 7 digits), 

 the six expressions v 2 mv, v 2 nv, v 2 pv are all squares when v = 65 2 . 



D. S. Hart 5 found three squares such that each increased by its root shall 

 be a square. Let ax, bx, ex be the roots. Take a 2 x 2 +ax = m 2 x 2 . For the 

 resulting value of x, b 2 x 2 +bx and c 2 x 2 +cx are squares if a?b 2 a?b+abm 2 

 and a 2 c 2 a 3 c+acm 2 are squares. Since this is the case when m = a, set 

 m = a+n. Multiply the resulting expressions by c 2 and b 2 respectively. 

 Then shall 



abc 2 n 2 +2a?bc 2 n+a 2 b-c 2 = D =A 2 , ab 2 c 



ir The Math. Repository (ed., Leybourn), London, 3, 1804, 97. The Gentleman's Math. 

 Companion, London, 3, No. 14, 1811, 300-2. Same in Math. Quest. Educ. Tunes, 

 14, 1871, 54; 24, 1876, 28. 



2 The Gentleman's Math. Companion, 5, No. 24, 1821, 59-60; 5, No. 26, 1823, 214. 



3 Math. Quest. Educ. Times, 14, 1871, 55-6. 



4 The Analyst, Des Moines, 2, 1875, 46-9. 

 6 Ibid., 3, 1876, 81-3. 



491 



