172 



1 f therefore 



PROFESSOR CHRYSTAL ON THE EQUIVALENCE OF 



{11} {22} {nn} | 



(25) 



denote the reciprocal matrix to | (11) (22) (nn) |, we have from the first n-\ 



equations of (24) 



k x :ko: :Kn={ln}':{2n}': . . . .{nn}', 



wliere {1^'j {nn}' are the set of relatively prime integral functions of D which 



arise by dividing {in} {nn} by their G.C.M., G n say. Therefore 



K 1 = X{ln}', /c 2 = \{2«-}', , K n = \{nn}', 



wliere X is some integral function of D, or a constant. 



Now, since | i;^ K n \ must be constant, (12) and (18) being equivalent, it 



follows that X must be a constant ; for | ^ x n 2 K n | obviously contains the factor 



X. We must therefore have 



[nn] = \[(ln){lnY+(2n){2nY+. . . +(nn){nn}'] = \ \ (11)(22) . . . nn\/G n . (26), 



that is to say, we must have, to a constant factor, [nn] = K/G jl5 where K is the char- 

 acteristic determinant of the original system. 



We are thus led to the following general rule : — 



Form the schemes 



u 2 

 u„ 



(11) (12) (In) 



(21) (22) (2n) 



(ril) (n2) (nn) 



(27), 



and 



<J\ 9-2 

 1 9 



(Jn 



{11} {12} {In} 



{21} {22} {nn} 



{nl} {n2\ {nn} 



(28), 



G l (x 2 G„ 



by means of the matrix of the given system and its reciprocal matrix, g l , g 2 , , g nl 



«ncZG 1 ,G 2 , , G n being the G.C.M.s of the constituents of the respective columns, 



then 



!J\ > 9l i ' On j 



K/G x , K/G 2 ,....., K/G ;l 



(29) 



are the diagonal coefficients of the variables x^ , x. 2 , , x n when these are first 



and last in diagonal order respectively. 



