(141) 
imate its true end. The distance estimated was small, how- 
ever. In Zrifolium the lengths were very well defined on 
account of the obtuse ends of the leaflets, but great care was 
necessary in getting the angular positions because of the great 
mobility of the leaflet-petioles. In Lugcnws no special diffi- 
culties were encountered. 
The Tabulation of Data. — After the measurements are 
taken we have for each species two groups of figures, one of 
the length and position of the leaflets in the normal series and 
the other of the length and position in the operated series. 
Putting these in parallel columns we may determine the 
normal average length and average position of each leaflet 
and likewise the operated average length and average posi- 
tion. 
As stated before, assuming that the normal averages repre- 
sent the positions and lengths which the operated leaflets 
would have had but for the removal of one of their number, 
we may conclude that the difference between the operated 
average and the normal average in each case is a measure 
of the effect of the removal. 
In order to indicate in some degree the trustworthiness of 
the results the probable error of the mean was calculated 
according to the well-known formula 
ca 
Probable Error of the Mean (P.E.M.) = 
6745 C¥v,—v;? 
vn ; 
In cases where the choice of specimens is made at random, 
the validity of the results is taken for granted when the dif- 
ference between the two averages is greater than the sum of 
their probable errors of the mean. In each of the present 
experiments, however, there was undoubtedly an unconscious 
selection as regards the length and position of the leaflets, 
due to the conscious attempt to make the two series similar at 
the start. But this factor increases the trustworthiness of the 
results on both sides, for the tollowing reasons: First, any 
* For the development of the formula see the text-books on the method 
of least squares. 
