492 
NOTE ON KIEPERt’s ¿-EQUATIONS. 
Moreover 
1 U 
, _ U id 11— U 2 — 1 
1 v? 1 — id v? (1 + u 2 + id) ’ 
id v 
u 
and thence 
1 (l M 4 
that is, 
V li 8 / 1 (w 8 -l)* 1 1 
iC (1 — u 8 )^ M id (1 — u 8 )^ M idM ’ 
L 2 = 1 + u 2 + u~ 2 . 
But we have 
u* = - 3 + 2 V2, 
and thence 
and 
w 2 = i (1 — V2), it -2 = i (1 + V2), 
Id = 1 + 2'i, 
whence 
L 12 + 10 L 6 - 
- 12 7 . 2 X 2 + 5 = (117 + 44 i) + 10 (- 11 - 2 i) -12 (1 + 2i)+ 5 = 0, 
or the ¿-equation is satisfied. 
Cambridge, 14 March 1887.