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},