688 HISTORY OF THE THEORY OP NUMBERS. [CHAP, xxm 



used in 



a_ a o+l b 1 



b~a+l'a+2' ~b~' 

 Set 



a _p u 



whence 



aq 



. 

 op aq 



Consider the case bpaq = l and let p = a, g = /3 be the least solutions. 

 Then 



a a a/3 



Proceed similarly with a/0. Many numerical examples are given. 

 A. Padoa 117 noted the equivalence of 



n+l x+1 y+l 



-=- -, (xn)(yn)=n(n-\-i). 

 n x y 



Hence if n is given we obtain all couples x, y by finding all pairs of positive 

 integers whose product is n(n+l), and adding n to each factor. 

 J. E. A. Steggall 118 found positive integral solutions of 



x+1 y+l_z+l 



by noting that xy must be divisible by x+y-\-l =a, and hence x(x+l) by a. 

 Hence for any integer x, determine a factor a>x+l of a; (x+1); then 

 y = a x 1, while z = x b where 6 = x(x+l)/a. T. W. Chaundy (pp. 74-5) 

 deduced (xz)(yz)=z(z+\} and set z = pq, xz = piq, where p, pi are 

 relatively prime . Hence yz = pqi, piq\ = pq-{-l. 



G. Ascoh and P. Niewenglowski 119 gave solutions of (1). 



A. M. Legendre 120 evaluated, up to 10 = 1229, 



2 4 6 10 w-l 

 3'5'7'll'" w ' 



OPTIC FORMULA ~H = -; GENERALIZATION. 



x y a 



An anonymous writer 121 noted that, if three regular polygons of x, y, z 

 sides fill the space about a point, then l/x+l/2/+l/2=l/2. If there are 

 four regular polygons of x, y, z, z sides, then l/x+l/j/+ 2/2 = 1. The 

 number of solutions is found, also for 5 or 6 polygons. 



117 L'interme'diaire des math., 10, 1903, 30-31. 



118 Math. Quest. Educ. Times, (2), 20, 1911, 50-1. 

 119 Supplem. al Periodico di Mat., 14, 1911, 101-4, 116-7. 



120 Throne des nombres, ed. 2, 1808; ed. 3, 1830. Table IX. 



121 Ladies' Diary, 1785, 40-1, Quest. 829; Leybourn's M. Quest. L. D., 2, 1817, 132-3. 



