CHAP. IX] RELATIONS BETWEEN SQUARES. 319 



in n-space). By multiplying a, by Va] for i = 2, , n, we get 



(a 2 + 2a^) 2 = (- a 2 + Za t -a*) 2 + 2a l (2aa i ) 2 , 



a formula noted by G. Candido. 66 



M. Moureaux 67 noted that successive applications of Aida's formula 

 gives 



M + + ay = bi + . + bi 



J. Cunliffe 68 noted that we can find any number of rational squares 

 whose sum is a rational square since n + k 2 = D, k = (4r 2 ri)/(4ar). 

 Thus, if w = 1 + 4 + 9 + 16, take r = 3, whence /b = 1/2, and we have five 

 squares whose sum is a square. 



L. Calzolari 69 found special solutions of 



^ + + ** = 2/ 2 

 by setting x { = k + a,-, y = /b + Za,-. The new equation is linear in each a,-. 



E. Lucas 70 noted that the sum of x consecutive squares may be a square 

 for x = 2, 11, 23, 24, but for no further value 1 < x ^= 24; the sum of n 

 consecutive odd squares is =|= D if 1 < n < 16. Cf. papers 76, 81, 86, 87, 100, 

 and 103 below; also papers 80, 130-8 of Ch. I; and Brocard 92 of Ch. XXIII. 



H. S. Monck 71 noted that V = (a 2 + 6 2 ) 2 = (2a6) 2 + (2bc + c 2 ) 2 if 

 a = b + c. Hence if 



a 2 = c\ + + ci|, t 2 = 46 2 c* + . + 4& 2 c* + (2bc + c 2 ) 2 

 is a sum of n + 1 squares. Also, 72 



Va] = {2s + (n + l)a} 2 , s = Sc,-, a,- = 2s + 2a - (n - l)c f . 



F. P. Ruffini 73 discussed the positive integral solutions i r ^ i r -i ^ ^ i\ 

 of 



il+ - +i* r = u, ii + + ir = v. 



Let Xi be the number of i's with the value 1, and x z the number with the 

 value 2. Set s = r x\ 2 - Then 



Xi + 4z 2 + 2i 2 = w, aji + 2x 2 + Si = v (3 ^ i, ^ i,_i ^ i\). 

 Solve for a?i and a: 2 , and require that the values be ^ 0. By x 1} 



il - 2i s ^ V = u - - 2v - Si 2 + 2Si, 

 where the summations extend over s 1 values of i. Hence 



The condition 1 + F ^ is treated similarly, first solving for i s _!. For 



66 Suppl. al Periodico di Mat., 19, 1916, 97-100. Case a r = r by Aida 148 of Ch. XIII. 



67 Comptes Rendus Paris, 118, 1894, 700-1. 



68 The Gentleman's Math. Companion, London, 3, No. 14, 1811, 281-2. 



69 Giornale di Mat., 7, 1869, 313. Cf. Ch. XIII. 123 



70 Recherches sur 1'analyse indetermine'e, Moulins, 1873, 91. Extract from Bull. Soc. 



d'Emulation Dept. de 1'Allier, Sc. Bell. Lettres, 12, 1873, 530. 



71 Math. Quest. Educ. Times, 20, 1874, 83-4. 



72 Ibid., 30, 1879,37-8. 



73 Mem. Accad. Sc. Istituto Bologna, 9, 1878, 199-215. Simpler than his paper, ibid., 8, 1877. 



