320 HISTORY OF THE THEORY OF NUMBERS. [CHAP, ix 



u n 2 1, v = 3(n 1), the initial pair of equations are the conditions 

 on a Cremona transformation. For u = n 2 2, v = 3n + 2p 4, they 

 are the conditions on the transformation of R. De Paolis, Mem. Accad. 

 Lincei, 1877-8. 



J. W. L. Glaisher 74 expressed the sum S(a,- a/) 2 of n(n l)/2 squares 

 as 



V 



where v = (n l)/2 or n/2 1, according as n is odd or even, and 



c m = cos (2mir/w), s m = sin (2mir/ri). 



G. Dostor 75 desired 2n + 1 consecutive integers such that the sum of 

 the squares of the first n + 1 of them equals that of the last n, and proved 

 that the first of the numbers is n(2n + 1) or n. 



A. Martin 76 proved for n = 3, 4, 5 that a sum of n consecutive squares is 

 not a square. Call x 2 the middle square when n = 3 or 5; the problem 

 reduces to the fact that 3x 2 + 2 = D or 5(x 2 -f 2) = D is impossible. 



G. Dostor 77 noted that, if a\ + + a n = np/2, 





! + +: = Z (P - a;) 2 , ;++ a*-! = P 2 + Z (P - i) 2 , 



i=l i=l 



the last by setting a n = 0, so that 78 a sum of n or ?i 1 squares is ex- 

 pressed as a sum of n squares. Also 



D. S. Hart 79 found squares whose sum is a square by subtracting 

 (s + m) 2 s 2 from I 2 + 2 2 + + n 2 and, by trial, expressing the dif- 

 ference as a sum of squares, which are then deleted from the n squares. 



J. A. Gray 80 noted that we may start with a sum S of squares, choose a 

 divisor a of S and set S + x 2 = (x + a) 2 , whence 2x = S/a a. 



Hart 81 considered the sum S of the squares of 2n 1 consecutive 

 numbers the middle one of which is x and, for special values = 181 of n, 

 made S a square. Cf. Lucas 70 . 



E. Catalan 82 proved there is a number equal to a sum of p squares and 

 having its square equal to a sum of 2p squares, by use of the identity 



(x 



zn 



74 Messenger Math., 8, 1878-9, p. 48. 



76 Archiv Math. Phys., 64, 1879, 350-2. Cf. Zeitschr. Math. Naturw. Unterricht, 12, 1881, 

 269; E. Collignon, Assoc. frang. av. sc., 25, II, 1896, 17; Ceearo 163 of Ch. I. 



76 Math. Visitor, 1, 1880, 156. Cf. Lucas. 70 



77 Archiv Math. Phys., 67, 1882, 265-8. 



78 For n = 3, E. Catalan, Nouv. Corresp. Math., 4, 1878, 3. 



79 Math. Magazine, 1, 1882-4, 8-9. 



80 Ibid., 76. 



81 Ibid., 119-122; errata corrected by Martin, 2, 1892, 94. 



82 Mathesis, 3, 1883, 199. 



