670] 
240 
670. 
[NOTE ON ME MUIR’S SOLUTION OF A “PROBLEM OF 
ARRANGEMENT.”] 
[From the Proceedings of the Royal Society of Edinburgh, t. ix. (1878), pp. 388—391.] 
The investigation may be carried further: writing for shortness u 3 , u 4 , &c., in place 
of (3), "'F (4), &c., the equations are 
u 3 = 1, 
w 4 = 2 u 3 , 
Hence assuming 
we have 
u 5 = 3 w 4 + 6u 3 + 1, 
u 6 = 4 u- 0 + 8 u 4 + 12 u 3 , 
u 7 = ou 3 + 10w 5 + 1om 4 + 18w 3 + 1. 
U — U 3 + U A X + U 5 X 2 + uyt? + U 7 X 4 + ..., 
w = - - + u 3 (2x-\- 6x 2 + 12a? 3 + 18a? 4 + ...) 
1 — x 2 
+ w 4 (3# 2 + 8x? + 15a? 4 + 22a? 5 + ...) 
+ u 5 (4a? 3 + 10a? 4 + 18x? + 26a? 6 + ...) 
+ u 6 (5x 4 + 12a? 5 + 21a? 6 + 30a? 7 +...); 
so that, forming the equation 
, x 2 
u 
u A { x 2 + 2x 3 + 
+ u 5 (2a? 3 + 4a? 4 + 
+ u 6 (3a? 4 + 6x 5 + 
3X 4 + 4a? 5 + ...) 
6x B + 8a? 6 +...) 
9a? 6 + 12a? 7 +...),