189 
of Edinburgh, Session 1880 - 81 . 
Consequently, of course, 
TJn-l “ U n . 3 = (» - 3)U w _2 + (271 - 6) U n _ 3 
+ (2n - 4)(U m _4 + U„_5+ • • • • +U"s)j 
and thus we are enabled to eliminate XJ„„ 4 + U W _ 5 + ... + U 3 . 
The result of doing so is 
(n - 2 )(U B - U w _ 2 ) - (72 - l)^., - U b . 8 ) 
= (n - 2)*U B _ 1 + (w - 2)(2» - 4)) U B . a + (2w - 2)(w - 2)> U„_ 3 , 
- (?z -=■ l)(?z ~ 3) ) - (n - l)(2?z - 6)j 
or 
(« - 2)U W = (tz 2 - 3?z + 3)U b _! + (zz 2 - 3zz + 3)U w _ 2 + (zz - 1)U m _3 (3). 
Putting (n - 1) for n we have also 
(zz - 3)U W=1 = (zz* - 5zz + 7)U„_ 2 + (n* - 5 n + 7)U W _ 3 + (» - 2)U n _ 4 , 
and therefore from this and (3) by subtraction 
(n - 2)U W = (ti 2 - 2w)Un-i + (2w - 4)U W _2 - (?z 2 - 6w + 8)U W _3 - (n - 2 )U W _4 , 
or U M = zzU w _ 1 + 2U w _ 2 -(zz~ 4)U n „3-U w=4 . . . . (4). 
Further, partitioning the term 2U W _ 2 into — U w _ 2 + ^-~- o -U w . 2 , 
71 — A 71 ” a 
this may be written in the form 
ZZ zz ~ 4 
XJ W - ZzU^.j 
or, say, 
whence 
or 
n Uw -2 C"n-2 71 — 2 TX„ _ o ' " . U 
zz - 2 7i - A 
n - 2 
-) 
Y 
^17. * ' 
n- 2 
w-2 ? 
Y = 
" 17 . 
zz — 4 
ZZ - 
6 
ZZ - 
8 
2 y 
4 45 
71 - 2 * 
ZZ - 
4 ' 
71 - 
6 .... 
zz- 4 
n - 
6 
n - 
8 
— V 
5 5J 
ZZ - 2 
n - 
4 
71 - 
6 
according as n is even or odd. But 
V 4 = U 4 -4U 3 -2U 2 = -2, 
V 5 = U 6 -5U 4 -|lT 3 = |. 
and 
