310 



SCIENCE. 



[N. S. Vol. XXII. No. 55S. 



Thus the formula assumes that the succes- 

 sive observations upon the same individual, or 

 t^ and t„, are constituents of the same norm in 

 conformity to the exponential law. The 

 method is simply a way of eliminating v from 

 the expression. The assumption upon which 

 the whole rests is that \_pq\ will be constant 

 no matter what the magnitude of v. 



Now, in tests as taken in psychological 

 laboratories it is evident that some change in 

 the norm results with each successive trial. 

 In all tests we find practise, warming-up, etc., 

 as factors leading us to consider the successive 

 i^'s as ordinates of a curve whose abscissa repre- 

 sents units of time. In practise, at least, we 

 assume a type of curve toward which all indi- 

 viduals tend. The individual curves are 

 variants in the group defined by the type curve. 

 The same conditions are found in growth of 

 stature. It has been shown that individuals 

 tend to the same curve of growth. Also that 

 when an individual varies greatly from the 

 type curve at one period of time he is likely 

 to vary less at another.' This implies that the 

 individual tends to fill a space type in a time 

 type so far as his physical growth is concerned. 

 If this is true we should expect to find the 

 same relations with respect to the form of any 

 organ. 



I have at hand data for measurements of 

 the alveolar arch with reference to the median 

 plane of the body. These measurements were 

 taken as ordinates of th^ curve as defined by 

 the teeth. This gives us the same general 

 geometric conditions as were found in curves 

 of growth. Now, before going on with the 

 data let us consider the problem as that of a 

 correlation between the ordinates of individual 

 curves, varying from the type curve according 

 to the exponential law. 



We may assume that OM represents any 

 curve. Let /\ be the first observed trial or 

 ordinate; t,, the second. Now, no matter 

 what the value of ^ may be, there must be a 

 correlation between 4 and t„ because of their 

 geometric relations. Also, there may be a 

 relation between {t^ — t^ and t„. 



Let 



- Franz Boas, Report of the Commissioner of 

 Education for 1896-7. 



[^1 — [^J = \P^, 



<^i! 0-2, 0-3, the respective variabilities of these 

 averages, 



a; = the deviations of t^ from [^2], 

 2/ = the deviations of ^1 from [tj, 

 y ^ 00 -\- p. 



Then (x -\- p) is a variable agreeing with a 

 variable y; p is the deviation of [P]. 



[xy'i = [o^'\ + [xp], 

 [rl = [^] +2[a?2J] + [f-]. 



Also 



bf] 



Hence, [xp'] should be zero, but this can 

 occur only when there is no correlation be- 

 tween t„ and P. 



The value of p may be expressed as 



[ir'\ = Uf]—2[wy-\ + lx']. 



Thus, the various terms of these equations 

 can be calculated from the data for the corre- 

 lation of t^ and t^. 



1 have calculated the following for the suc- 

 cessive ordinates of the alveolar arch : 



In this table I have correlated the smallest 

 dimension with each of the successively large 

 dimensions as indicated by t. [a;^] is taken as 

 a constant; 0-3 appears to be constant. The 

 correlation of t ^^_^ and P varies for different 

 points in the curve OM, in such manner that 

 two points may be found for which the corre- 

 lation is zero. \_xp] varies with the magni- 

 tude of [P], which in this curve is dependent 

 upon the distance between ^^ and t,. 



I have no good data for a psychophysical 

 test. The only available data, at this writing, 

 are the reaction times in the Columbia Uni- 

 versity tests. To preserve the form of the 

 preceding table the fifth reaction was taken 

 as the point of departure. Thus : 



