Chap. XXIII] 
Optic Formula l/x+l/y = lfa; Generalization. 689 
D. André 122 deduced x—a=d, y—a = e, where de —a 2 , the pair of divisors 
d=e=—a of a 2 being excluded. Züge 123 gave x = a+p 2 , y = a+q 2 , where 
pq = a. F. Schilling 124 noted that Ztige’s solution is incomplete and gave 
that due to André with a geometrical interpretation of the optic formula. 
A. Thorin 125 asked if l/a = l/ai+l/a 2 has integral solutions besides 
a = mn, ai = m{n+l), a 2 = mn{n-\-1). 
A. Palmström, J. Sadier, and C. Moreau 126 each gave the solution 
a = \mn, 
and noted that 
(1) 
ai = Xm(m+n), a 2 = \n{m+n), 
CL CL\ CL n 
has the special solution 
a=Xcki • • • a n , di = Xscki, 
*, CLn Xsoi, 
•-E 
«1* * 'Gin 
Dujardin 127 stated that, if n = 2, all solutions are given by 
a 2 -a+\, ai = a+-~ (X a divisor of a 2 ), 
A 
while (1) may be written Aa n = a{Ba n -\-C), where and B, C 
are integral functions of a ly • • •, a n _ x [with C=A]. Then Ba=A—AC/\, 
where \ = Ba n J rC. Hence give to a h • • *, <2 n _i any values and choose a 
divisor X of AC. 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 21 ¡xt 
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 
- = -+• 
X1 X 2 X n 
which may be written in the form 
—x 2 
x 3 
0 
0 • 
•• 0 
—x 2 
0 
Xi 
0 • 
•• 0 
- 
—x 2 
0 
0 
x 5 • 
•• 0 
= 0. 
—x 2 
0 
0 
0 • 
• Xn 
Xi—X 2 
Xi 
Xx 
Xi • 
• x x 
122 Nouv. Ann. Math., (2), 10, 1871, 298. 
123 Zeitschrift Math. Naturw. Unterricht, 26, 1895, 15-16. 
124 Ihid., 491-3. 
125 L’intermédiaire 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. KI., 
No. 7 (German). L’intermédiaire des math,, 5, 1898, 81-3. 
45