116 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
(12) This desired equality I have found to be 
{[a\x+ \J)]y} n+l — {{a - n)x + {b - n)y}{\a\x + [b]y} n - nxy{a + b - n + l){[«]a? + [b]y} n ~ 1 
so that if we put F n for {[ct]x-\-[b]y} n we have 
F ^{ax + by} =0 
F 2 — {(a — V)x + (b— l)?/}F 1 + ^(a + 6) =0 
F 3 — { {a — 2 )x + ( b — 2 )y }Fg 4- 2 xy{a -\-b — 1)1 1 = 0 
F 4 — {(a — S)x+ ( b - 3)//}F 3 + 3 xy(a + b— 2)F 2 = 0 
and thence 
{[«> + [%}“ = 
ax + by xy . . ... 
a + b {a— V)x + (b— l)y 2 xy . . . 
a + b — 1 (a- 2)x+ (b— 2)y 3 xy . . . 
a + b — 2 {a — 3)« + (6 — 3)y . . . 
which, besides being ideally simple in its law of formation, remains 
invariable to any and all of the above-mentioned changes in a , x , b , y . 
Rondebosch, S.A., 
1 6th February 1921. 
{Issued separately August 23, 1921.) 
