366 Mr. F. Y. Edgeworth on the 
Another variant in this and, be it said once for all, all the 
subsequent typical problems, is when the observations stand 
for, not the simple value of, but definite functions of, the real 
quantity. 
I. A (2) Griven a set of observations # 1? a 2 , &c, and given 
that they were generated by divergence according to a given 
curve of probability of which the modulus is not given but 
sought. Let c be sought modulus ; then it must be taken so 
— ) e & should be a maximum; whence 
^_&£, 
it • 
& n 
A variant would be the case when the observations are dis- 
tributed into groups, each with a different modulus, and the 
ratio between the moduli is given. 
I. A (3) By the degradation of the data in either of 
the preceding problems we reach the complex problem : — 
Given a set of observations « 1} «r 2 , &c, and given that they 
have been generated by divergence according to one and the 
same probability-curve from a single point, but given neither 
that point nor the modulus, to find both. Put as the proba- 
bility of the concurrence of a particular central point and a 
particular modulus 
Fdgdc 
where 
U 
YdZdc 
2 /l\ n r ^i-g) 2 -[-(^2-r 2 +&c. ] 
7T» X P= (I X6 L c2 J. 
Equating to zero the first term of variation (that is, equating 
to zero the coefficient of di; and also the coefficient of dc in 
the development of the above proposition), we have two equa- 
tions to determine £ and c. Whence the most probable value 
of f is the mean of x x , x 2 , &c, and the most probable value of 
c is the square root of twice the mean square of apparent errors, 
viz. 
-(g-*i) 8 + (g-* a ) a + &c- -|. 
a solution of the compound problem which follows the analogy 
of the simple problems (1) and (2), and which holds whether 
the number of observations be finite or infinite. 
I believe that this conclusion is usually restricted to the 
* Cf. Merriman, ' On Least Squares,' p. 143. 
\Ap 
