42 G. H. KNIBBS. 



expressed, will of course be unity; so that whether k- 1 ratios,, 

 or k absolute values are assigned, the result is the same, viz., that 

 there are m linear equations in m variables. These k terms are 

 then known terms, or more correctly, terms that can be numerically 

 evaluated ; they may therefore be summed and denoted by k 1? k 2 , 

 etc. Thus (8) takes the form 



a k a + 6 k /3+ +x k = (14) 



in which k has all values from 1 to m. Hence, as before, writing 

 A for the determinant of the mth order, whose constituents are the 

 coefficients of the quantities to be determined ; and A 1? A 2 , etc. 

 for the determinants derived therefrom by substituting the k or 

 absolute column for the first and second columns etc., we have 

 again 



*--%"> p = ~x ; etc (15) 



in which a, /3, etc. may be considered as ratios merely, whether 

 k - 1 ratios — one of the others being taken as unity — or k weight- 

 coefficients, have been assigned. 



7. Position of a single section. — Suppose the original function 



to be 



A Z = A+Bz>, (16) 



then either directly, or by propositions {a) (b) and (c), it is evident 

 that a single value of the variable, or " section ' n may be taken at 

 which the value of the function shall be a mean, (i.e, ordinate or 

 area). Thus : — 



Prop. (d). If the original function have one positive index only 

 a single mean value way be taken at a point on the axis determined 

 by the index. 



The position of the point is found as follows : — 



or logarithmically 



a= W^T) (17) 



1 The weight-coefficient a will of course he unity. The " section " i& 

 the point on the axis where the value of the function is a mean value, in 

 this case. 



