176 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Because the right member of (24) is linear in the p' 's and pj has the 

 coefficient 1 iu the expression for 8jo,„ but does not occur in the expres- 

 sion of any 8p with suffix less than w, pj can be expressed as a linear 

 function of the 8;/s with suffixes not greater than m whose coefficients 

 are polynomials in the /^'s. The expression for p^' in terms of ^p.2, 

 hpz, ^Pi, and Bp^ shows that the coefficients of the S/»'s in the expres- 

 sions of the p' 's are not generally linear. If we substitute the expressions 

 of the jo''s as linear functions of the 8jo's in (22) we obtain S\j/ ($) as a 

 linear function of the 8/»'s whose coefficients are polynomials in the p's 

 multiplied by derivatives of if/d) with respect to $; therefore, i/^ (^) is a 

 function of i and the p's with suffixes as great as 2 alone, say * 



(25) yl'{^) = F{^_,p^,p^,pi,p,,...). 



This result confirms the statement made above that all theory can 

 determine is the probability that the residual shall lie between given 

 limits, 



V. Change of the Unit of Measurement. 



If we reduce the unit of measurement in the ratio 1 : A, the numerical 

 values of all measurements are multiplied by X, so that, for given obser- 

 vations, x, ar,), I, d^, pk{ioT 2 ^ k) have to be replaced hy \x, Xxq, X^, 

 X • dt, X}' Pi,, and, by assumption III, the probability that the residual 

 lies between X $ and X $ + X ' d^ with the new unit is the same as that 

 it lies between | and $ + d^ with the old unit ; therefore, 



F(X$, X>2, X>3, A>4, • • •) • ^d^=F($,p,,p,,p,, ...)■ d$, 

 that is 



(26) F(Xi, AV2, A^s, A>o • • •) = X ^(^'P"''P'' P*' • • •) 



for all values off, p^, ps, pi, • . . The relations (25) and (2G) express 

 that i/'(l) is a homogeneous function of $, \/p-2, VjOa, "Vpi, etc. of de- 

 gree — 1, that is, with the notation of (6) and (7), 



(27) ^(^) = lf(s, pg, P4,P5,...)5 



for convenience, we shall sometimes abbreviate ij/ (i) to ij/ and 

 /(«j Ps> Pi' P5' • • •) to/(s) or/. 



* This conclusion implies that every infinitesimal change in yf/{^), for a given 

 value of |, can be effected by the addition of an infinitesimal source of error, — 

 which ought, perhaps, to be stated as a fourth assumption lying at the base of the 

 general theory. 



