Student 
271 
see no general way of proving the results I propose merely to state my results as 
follows : 
(1) A tie of t on one side which is opposed by no ties on the other side will 
24?i 
diminish ^ bv ^-^tn, — ^ if t be odd and by , if t be even. 
6 
(2) Overlapping ties on opposite sides interfere with the above simple rule, the 
?r — 1 . . 
total to be subtracted from — g- being increased or decreased according to Table IV. 
TABLE IV. 
Distance, between centres of ties * 
.(■ tie 
;/ tie 
0 
J 
2 
1 
2 
3 
4 
5 
Si 
6 
2 
2 
+ 1 
0^ 
.3 
3 
2 
3 
+ 2 
0^ 
-1 
0^ 
4 
2 
+ 2 
0-^ 
4 
3 
+ 1 
-1 
0^ 
4 
4 
+ 6 
0 
-1 
0-* 
5 
2 
0^ 
5 
3 
+ 4 
- 1 
-1 
0-* 
5 
4 
+ 3 
- 2 
- 1 
0^ 
5 
5 
+ 10 
0 
-4 
- 1 
0^ 
6 
2 
+ 3 
0-* 
6 
3 
+ 2 
- 1 
-1 
0^ 
6 
4 
+ 10 
+ 1 
_ 2 
- 1 
6 
5 
+ 7 
-2 
-4 
-1 
0^ 
6 
6 
+ 19 
+ 4 
-4 
- 4 
-1 
0^ 
7 
2 
0^ 
7 
3 
+ 6 
-1 
- 1 
- 1 
0^ 
7 
4 
+ 5 
- 2 
-2 
-1 
0-- 
7 
5 
+ 16 
+ 2 
-5 
- -4 
-1 
0-* 
7 
6 
+ 13 
- 1 
— 7 
-4 
- 1 
0^ 
7 
7 
+ 28 
+ 7 
-6 
- 10 
-4 
-1 
0-^ 
etc., etc. 
As an example of the use of Table IV, suppose a set of eight ranks to contain 
on the X side a tie of 5 centred at 3, i.e. let the ranks be 3, 3, 3, 3, 3, 6, 7, 8, and 
let the y ranks have a tie of 4 centred at 2^, i.e. let the y ranks be 21, 2^, 2i, 2|^, 
5, 6, 7, 8. Then the amount to be subti-acted from — ^ — is firstly !^ (for the 5 tie) 
+ 1 (for the 4 tie) + 5 (from Table IV) = 11. Had the y ranks been 1, 3^, 3J , 31, 
3|, 6, 7, 8, the correction would be the same, but if the y ranks were 1, 2, 4^-, 4-| , 
4^, 4^, 7, 8, the correction would be § + f - -5 = f , 
and if 1,2,3,51,51,51,5^,8 A + 1-1 = I, 
and if 1,2,3,4,61,61,61,61 t + § = i. 
It is only with very small and much tied samples that the correction is appre- 
ciable. 
