127 



Further; of the two cases a and b it will be sufficient 

 to consider only the first. For if (case b) any quantities z, 

 diminishing with increasing x, are normally distributed, 

 then the quantities — z, which belong under case a, will 

 also be normally distributed ^). 



Suppose, therefore. the z to increase regularly with 

 the X and let x^ and Zj be two corresponding values and 

 let it be remembered that as each individual x must hâve 

 its corresponding individual z we must suppose the quan- 

 tities X and z to be in equal number. 



It folio ws that to any x below Xi corresponds a value 

 of z below Zi and to any value of x in excess of Xi a 

 value of z in excess of Zj. For, if to any value Xq below 

 X] corresponded a value Zq exceeding z^, we would hâve, 

 corresponding with the increase Xi — Xq of x, the decrease 

 z^ — Zq of 2' which is contrary to our supposition. 



As therefore ail the x below x^, and no others, hâve 

 their corresponding values of z below z^, we conclude: 

 that the total number of z below z^ is equal to the total 

 number of x below x^. 



Remembering the meaning of the scheme, we may 

 express this by saying; if certain quantities z are pure 

 functions of the quantities x, then those values of x and 

 z will correspond which in their frequency-schemes hâve 

 equal ordinates. 



This being granted, let in fig. 8 on the left hand side 



M More generally: if any quantities z are normally distributed then 

 it must be évident that 2 times, 3 times . . . . b times there quantities 

 must be similarly distributed. Also that this distribution remains normal 

 if we increase ail our quantities by the same amount a. This then 

 cornes to saying that: if the z are normally distributed: the quantities 

 a -{- bz (where a and b may either be positive or négative) are also 

 normally distributed. In reality therefore our problem must be considered 

 to havc an infinity of solutions. It is sufficient however to find one. 

 From which we may, if we like, form ail the others. 



Recueil des trav. bot. Néerl. Vol. XIII. 1916. 9 



