OF LOGICAL CLASS-FREQUENCIES, ETC. 
] II 
Therefore the limits to — cos 2z7r are 
cos 2x7r . cos 2'?/77 it v/1 + cos^ 2.r7r cos'2^/77 — cos^ 2a;7r — cos®2_y7r 
= cos 2x77 . cos 2/y77 sin 2x77 sin 21 /- 
= cos {x :T: y) 2 tt. 
Also 
— cos 2277 = cos (z i 4) 277. 
Therefore the limiting values to z are given by 
± i ± {x ± y). 
Here we need not take all three signs positive, for l)y the inferior conditions z 
cannot be greater than 0'5 ; nor all three negative, for x cannot be less than zero. 
Hence the limits given are 
0'5 + x — y ] 
0-5 — X + 7/ / 
0'5 — X — y ] 
— 0'5 + X -b 7/ f 
These are precisely the limits given by the conditions of consistence stated at the 
commencement of § IG. Thus the limits to one correlation coefficient in terms of tlie 
two others are most simply regarded as functions of the limits to the quadrant 
frequencies. The table below gives the limits to z for ditferent values of x and y 
the table is of course symmetrical with regard to x and y. 
TxVBLE showing the limits to z in terms of x and 7/, X = (AB)/(U), y = (AC)/(U), 
2 = (BC)/(U), for the case of Equality of Contraries. 
Value of //. 
Value of x. 
0. 
0-1. 
0-2. 
0-3. 
0-4. 
0-5. 
0 
0-5 
0-4 
0-3 
0-2 
0-1 
0 
0-1 
0-4 
0-5 
0-3 
0-4 
0-2 
i 
1 
1 CO T-H 
o o 
i 
0-2 
0 
0-1 
0-2 
0-.3 
0-4 
0-2 
0-5 
0-1 
0-4 
0 
CO 1 
1 o o 
1 
1 
0-2 
0-.3 
0-2 
0-3 
0-1 
0-4 
0 
0 -ij 
0-1 
0-4 
0-2 
0-3 
0-4 
0-1 
0-2 
0 
0-3 
0-1 
1 
O O 
lo hb- 
o o 
0-4 
0-5 
0 
0-1 
0-2 
0-3 
0-4 
0-5 
