rU<3FESS0R K. PEAE.SON ON THE MATHEMATICAL THEORY 
•2U0 
curve iu certain cases is a prubal^le description, in a fair number (_»f other cases it is 
a rougli approximation, in many it is impossible. We must then start from its simple 
axioms (a), (/3), (y), and generalise in the next simplest manner. We assume that our 
contributory cause-groups are not indefinitely great in numbei', hoAvever numerous are 
the causes which determine the contribution of the group, that this contribution is not 
ecpially likely to be positive or negative, and finally that the contributions of the cause- 
groups are not independent but correlated cjuantities.^^ The simplest extension of the 
theory of Gauss, Laplace, and Poisson in these directions leads us to the svstein of 
skew curves which have been applied in this memoir. 1 have treated them here purely 
from the experimental side. I have endeavoured to show in a fairly wide series of 
o1:)servatlons that the system of skew curves will, and the normal curve will not, 
satisfy the demands which we may fairly make on a theoretical frequency distriljution. 
In another paper I shall consider the philosophical points which have been raised by 
Eogwortii, Lipps, and other recent writers. My present object has been to show 
certain failures in the ordinary theory of errors, and especially in its application to 
personal equation, and to show how existing theory may be widened so as to describe 
obsei'vations within the limits of the probable errors of the constants determined on 
the basis of random sampling. 
12. Summary and Conclusions. 
If we attempt to sum up the results reached, their importance seems to rest on the 
amount of weight that is given to tlie experimental material. Can we look upon 
this as tyjDical of the measurements usually made by |)hysicists and astronomers I 1 
am unable myself to differentiate it, or to see causes for tlie higb correlation of 
judgments which are peculiar to our experiments, and not to observations such as are 
daily made in the physical laboratory or tlie obsei-vatoiy. If this be so, then we 
must conclude as folloAvs 
(o.) Tlie personal equation, while tending to a constant value, a})pears subject to 
fluctuations far exceeding those of random sampling. 
(h.) These fluctuations in the case of two or more oliservers, v hether dealing at 
the same time with the same phenomenon or measuring at different times the same 
* Siq)pose, for ex;ini))le, that tlie cause-groujjs were those series of incalculable causes ivhich deterniiiie 
(e) whether a coin shall fall head or tail uppermost; (//) whether an //-sided teetotum shall fall on one of p sides 
of one colour or not; (/) whether a card drawn out of a pack of nj) cards of j) suits is of any particular suit. 
Then, if an indefinitely large number of coins be thrown together, the frequency distriljution for heads 
satisfies all the fundamental axioms (a), (/d), (y) of the normal curve; if a hnite number of teetotums be 
spun and the sides of the ^/-colour counted, Ave have satisfied (y) only. If s cards be draAvn simultaneously 
from our pack and the cards of one suit counted, then Ave have satisfied no one of the three fundamental 
axioms; there is correlation betAA'ecn the contributions of the cause-groups. This is only a rough illustra¬ 
tion of the manner in which one or more of the fundamental axioms can be suspended artificially, but it is 
not Avithout suggestion for the processes of nature. 
