No. 615] THEORIES OF C BO 8 SIN O OVER 255 



and .031 is the same as that resulting from .989 and .969. 



2. If one of the two exchange ratios is changed to its 

 complement, the cross-over ratio is changed to its com- 

 plement. 



That is, if the cross-over ratio resulting from x and y 

 is C, the cross-over ratio resulting from x and 1 — y, or 

 y and 1 — x is 1 — C. 



For: 



x + (1 - y) - 2x(l -y)«l- (x + y- 2xy) 

 But the first member of this equation is the cross-over 

 ratio from x and 1 — y, while the second member is 1 

 minus the cross-over ratio from x and y. The same 

 result is reached if we take y and 1 — x. 



Thus, as the table shows, the cross-over ratio resulting 

 from .2 and .3 is .38, so that the cross-over ratio from .2 

 and .7 is .62, as is likewise the cross-over ratio from .8 and 

 .3 (.38 + .62 = 1). Similarly, the cross-over ratio of .011 

 and .031 is .0413; hence the cross-over ratio from .011 and 

 .969 is .9587. 



3. When the cross-over ratio is less than %, the ex- 

 change ratios x and y are either both greater than % or 

 both less than y 2 ; one can not be less than y 2 , the other 

 greater. That is : 



If C < y 2 then either x < V 2 and y < y 2 or x > V 2 and 

 y > y 2 . For let us suppose that x = V 2 — a and y = % + 

 b, in which a and b are any positive quantities. Then 

 C = x -f- y — 2xy = y 2 + 2ab. Therefore x can not be less 

 than V 2 and y more than V 2 . 



On the other hand, if x == V 2 — a and y = y 2 — b, or if 

 x = y> -f a, y = y> -+- b ; in either case C = x + y — 2xy = 

 y 2 — 2ab. So that in these cases the cross over-ratio C is 

 less than V 2 . 



4. Conversely to 3, when the exchange ratios x and y 

 are both less than V 2 , or when they are both more than V 2 , 

 the cross-over ratio is less than V 2 . 



That is, when x < V-i and y < y 2 . or when x > V 2 and 

 y > y 2 ; in either case C < V 2 . This was proved under 3. 



5. When the cross-over ratio is greater than V 2 , one ex- 



