Law of Error and Correlated- Averages. 431 



the case in which each member of a group is — not a linear 

 function — but some function of a linear function of numerous 

 elements oscillating in the manner defined above : say F(E); 

 where 



T>\, p>2, &C, are of the same order of magnitude : x^ .r 2 , &C. 

 oscillate respectively between limits which we may write 

 without loss of generality — a x , — a 2 , &c. ; while Sa may be 

 taken as the unit. Now by the usual theory E fluctuates 

 about its average value — say X — in conformity with a 



Probability-curve of which the modulus is of the order — — * 



sf n 



(Sa being =1). Put E = F(X + f) and, expanding, write 



F(B)=F(X) +fF'(X) + |F"(X+0f) 



(where 6 is a proper fraction). This expression is approxi- 

 mately equal to its first two terms F(X) + fF'(X) ; provided 

 that, for the values of F(E) with which we are concerned — 

 at most from to 1 (if Sa=l) — the function is free from 

 singularity and continuous in senses defined by the con- 

 dition that fF"(X + 0f) is small in relation to F'(X), for 



^ 1 



values of £ of the order — -=-t« 

 v n 



That condition holding, we may reason thus: — Of the 

 group formed by the varying values of E the greater part is, 

 by the usual theory, arranged according to a probability- 

 curve with centre X (the average value of S), up to a dis- 

 tance from that centre, +f, where £ is a small fraction. But 

 every value of F(X + f)=F(X)+£F'(X) nearly. Therefore 

 the greater part of the group F(S) ranges in conformity with 

 a probability-curve whose centre is F(X), and whose modulus 

 is that of f multiplied by F'(X). 



For example, let F(E)=S 2 ; where E is the sum of m 



* Above, p. 430, and the article there referred to. 



t This is a kind of assumption continually made, I think, by mathe- 

 maticians. To take an instance cognate to the present subject, Laplace, 

 when introducing the Method of Least Squares (Theorie analytique, 

 Book II. ch. iv. art. 20) supposes the datum of observation to be a func- 

 tion of the " element " which it is sought to determine. An approximate 

 value for this value having been obtained; and this value, plus a correc- 

 tion z having been substituted for the element ; " expanding in ascending 

 powers of z [en reduisant en sfrie par rapport a z"j and neglecting the 

 square of z, this function will take the form h-\~2 )Z " It will take that 

 form only upon the condition above stated. 



