CHAP, xxm] OPTIC FORMULA l/x+lfy=l/a; GENERALIZATION. 689 



D. Andre* 122 deduced x a = d,y a = e, where de = a z , the pair of divisors 

 d = e= a of a 2 being excluded. Ziige 123 gave x = a-{-p 2 , y = a+q 2 , where 

 pq = a. F. Schilhng 124 noted that Ziige's solution is incomplete and gave 

 that due to Andre" with a geometrical interpretation of the optic formula. 



A. Thorin 125 asked if l/a=l/Oi+l/a 2 has integral solutions besides 



a^^ tyWY} ft 4 ' " *VM I -V} I 1 \ fl m .1 > }'} >1 I *V7 I I I 



* * VI V \M 1 " ~~ II V \ IV I JL I Vt'V " ~~ 1 1 VI V\ IV I 1. / 



A. Palmstrom, J. Sadier, and C. Moreau 126 each gave the solution 



a=\mn, o 1 = Xm(m+n), 

 and noted that 



(D !!+... +1 



a ai a, 



has the special solution 



' '<X n , d\- 



=i a,- 

 Dujardin 127 stated that, if n = 2, all solutions are given by 



a 2 



a 2 = a+X, ai = a+ (X a divisor of a 2 ), 



X 



while (1) may be written Aa n = a(Ba n -\-C), where A = a\- -a n -\, and B, C 

 are integral functions of a i} , a n -i [with C=A]. Then Ba = AAC/\, 

 where \ = Ba n +C. Hence give to ai, , a n _i any values and choose a 

 divisor X of A C. Take as B and a two integers whose product is A AC/\. 

 If X C is divisible by B, we get a solution. 



M. Lagoutinsky 128 stated that if n = 3 the complete solution of (1) is 

 given by formulas involving 13 parameters. 



V. V. Bobynin 129 discussed the expressing of fractions in the form Sl/re* 

 in the papyrus of Akhmim (Achmim), about the seventh century, and in 

 the Liber Abbaci of Leonardo Pisano. 



A. Palmstrom 130 treated, as an example of a more general type, 27 



which may be written in the form 



122 Nouv. Ann. Math., (2), 10, 1871, 298. 



123 Zeitschrift Math. Naturw. Unterricht, 26, 1895, 15-16. 



124 Ibid., 491-3. 



125 L'interm&iiaire des math., 2, 1895, p. 3. 



126 Ibid., 299-302. 



127 Ibid., 3, 1896, 14. 



128 Ibid., 4, 1897, 175. 



129 Abh. Geschichte Math., IX, 1-13 (Suppl. Zeitsch. Math. Phys., 44, 1899). 

 130 Skrifter Udgivne af Videnskabsselskabet, Christiania, 1900 (1899), Math.-Naturw. Kl., 



No. 7 (German). L'intermediaire des math., 5, 1898, 81-3. 

 45 



