﻿out of the Theory of JRectifiahle Compound Logarithmic Waves. 377 



i. e. is the resultant of 



U, V, W, dt {x 2 - 1 2 ) {if - f) (z* - f) . 

 This enables us to see that the required resultant is the product 

 of all the resultants of the systems that can be formed by the in- 

 terchange of a } b, c after the pattern of the system 



(a±d)x +{b±d)y + (c±a)z, 



{a ± d)x 3 +(b± d)y 3 + (c± d)z 3 3 



{a±d)x r >+{b±d)y d +{c±d)z b 



(the signs in the coefficients of the same column being alike, but 

 independent as between column and column), multiplied by the 

 resultant of 



ax -\-by +cz, 



ax 3 + by 3 + cz 3 , 



ax 5 -f- by 5 + cz 5 , 

 multiplied by 



#1.3.5. 



and by continuing this process^it is obvious that the required 

 resultant will be made up exclusively of factors of the form 



d\ {d± c) "-, 7 (d± c ± b) \ (d±c±b± a)\ 



So in general for n equations, it may be shown in like man- 

 ner that the resultant is the product of factors of the form 



{a 1 ±a 2 ±a 3 ... ±a i ) n n,i, 



where u n>i is a function of n and i to be determined. But by 

 aid of the method of reduction above indicated, and fixing his 

 attention on those factors of the resultant only in which the 

 single coefficient retained in the substituted equation appears, 

 the intelligent reader will find no difficulty in ascertaining 



(1) that tt W|1 = 1.3.5...(2n-l), 



(2) that u nti =(i— ]>„_i, i_i. 



These two conditions furnish us with the following Table of 

 double entry : — 



i— 



= 1, 



% 



3, 4, 5, 6 



= 1 



1 







= 2 



1 



1 





= 3 



3 



1 



2 



=4 



15 



3 



2 6 



= 5 



105 



15 



6 6 24 



= 6 



945 



105 



30 18 24 120 



