31
Z"\
7,"
{« + /321(1), {« + /31} (2), {« + /30}(3),
{a + /3012} F = {/301} , +2 {/31} , +1
{ „ j Z" = {«01} , -2{«1} , +1 ;
{a + /334} (1), {a+ /314} (2), {a +/303} (3), {« + /301} (4);
{a +/301...4} W = {/3012} , +3{/312} , + 3 {/32} , +1
{«+/3456}(1), {a+/3156}(2), {«+/3036}(3), {«+/3015} (4), {a+/3012}(5);
{«+/301... 6)U = {/30123} , + 4 {/3123} , + 6 {/923} , + 4]/33}~ , +1
&c.
read a + /3. Y = /3 (1) + (2),
„ . Y'= a (1) — (2),
a + /3.a + /3 + l.a + /3 + 2.£’ = /3.^ + l.« + ^+2.(l) + 2./3+l.« + /3 + l.(2) + a + /3.(3),
„ „ ,, -Z =«.« + 1.« + ^ + 2.(1) + 2.a + 1.« + /3 +1.(2) +a + /3.(3),
&c.,
the law being obvious, except as regards the numbers which in the top lines occur
in connexion with a + /3 in the { } symbols. As regards these, we form them by
successive subtractions as shown by the diagrams
34
34
456
456
5678
5678 &c.
2
14
3
156
4
1678
11
03
12
036
13
0378
2
01
21
015
22
0158
3
012
31
0127
4
0123
and the statement of the result is now complete.
In part verification, starting from the Y-formulae (which are obtained at once),
assume
{« + /32} (1), {« + /31} (2), {a +/30} (3),
{a + /3012} Z =
{ » )*'-
Z" =
X"
we must have
that is,
(1) (2)
{« + /3012}. Z +Z' = {« + /3012} F, = {« + /312} ({/30}, + 1)
.Z' + Z" = { „ }F, ={ „ }({«0},-l)
{a + /32} . \ +V = {a + /312} {/30},
. X' + = { „ } {«0},