40 



G. H. KNIBBS. 



We proceed to the solution for the evaluation of these ratios. 



3. Determination of the ratio of the m + 1 weight-coefficients, 



when the number ra of indices is one less than the number of values 



of the variable. — Let the ratio of the weights be denoted by 



a =a/V; /?' = /?/v;... etc (9) 



then, since each term is divided by v, equations (8) may be written: 



a x a' + b x p' + . . . + n x = 0\ 

 a 2 a + 6 2 /3' + ... +n 2 = { 



a m a , + bj3'+...+n m = ) 

 in which m = n - 1 ; that is, there are in lines or rows, and m + 1 

 columns. Let the determinant of the rath order 



(10) 



a x 



6i ■ 



. m y 



a 2 



b 2 . 



. m 2 



. 



. 



. 



» m 



6 m . 



• ™ m 



(11), 



in which m is the coefBcient of the term, /x' say, in the rath or 

 (n- l)th column in (10); and let \, A 2) ... A m denote the deter- 

 minants derived from (11) by substituting, for the 1st, 2nd, and 

 rath or (n - l)th columns therein, the n or final column in (10): we 

 shall then have for the ratios of the weight-coefficients, v being 

 taken as unity, 



•'="---£; ?-£--$■; ..y=^ = -^;...(i2) 



v A v A v A 



provided that A is not 0, which since a, b, c, etc. and p, q, r, etc., 

 are both positive and in ascending order of magnitude may readily 

 be demonstrated as usually true. 



4. Number of indices greater than the number of values of the 

 variable, diminished by unity. — If (10) m = n + k, k+l equations 

 will exceed the number necessary to determine the ratio a', /3'etc; 

 hence, unless k + 1 suitable conditions are imposed, that number 

 of equations will be inconsistent. In general, therefore, the 

 number of different indices should not exceed n — 1. 



Since this last number of equations determine a', /3' etc.; and 

 since also a, b etc. are by hypothesis fixed quantities, the problem 



