408 THE AMERICAN NATURALIST [Vol. LYI 



will be proportional to 



and the likelihood of any given pair of values for and 

 P2 will be proportional to the same quantity. In order 

 to make this quantity a maximum for variations of 

 and p^, we have the equations 



These equations are exact, but for practical purposes 

 we need equations linear in p^ and p^, and a first approxi- 

 mation is sufficient; if p differs little from x/{x + ?/) ~ 

 x/n, then 



So that we may rewrite equations (1) in the practical 

 and approximate form 



For each percentage observation, therefore, we have 

 merely to calculate the two quantities n^/xy and n^/y-j 

 then normal equations may be constructed in the form 



where a^^ is the sum of the quantities n^/xy for which 

 both ^1 and p^ are involved, a^^ the corresponding sum for 

 all in which p^ is involved, and the sum of the quanti- 

 ties rv'/y for which pi is involved. 



3. Practical Example 

 In order to illustrate the practical application of this 

 method to a complex case, we will consider the location 

 of the 8 genes, from Eeduced to Eimmed, in the middle 



