﻿out of the Theory ofRectifiable Compound Logarithmic Waves. 379 



is zero. Hence the required resultant will contain the product 

 of the resultants of the 1.3.5 systems formed after the above 

 pattern ; and as this product will be of 1.3. 5 (1.3 + 1. 5 + 3. 5) 

 dimensions in the constants, it must be not merely contained in, 

 but identical with, the required resultant. Thus the new set of 

 functions regarded as hyper-loci (like the former set) can only 

 be made to intersect in one or another of a fixed group of points. 

 Moreover, passing to the case of 2n equations, it is obvious that 

 the resultant of such system will be made up exclusively of fac- 

 tors of the form 



(«i + « 2 + • • • +«;— a i+1 —ai. 



a 2i ) J n, 



* 



where J K}i is a function of n and i to be determined. The value 

 oiu n>i , which has been found above, leads to this without diffi- 

 culty. By an obvious method of calculation it may be shown that 



n . (n — 1) 



_^_ f2n . (2n-l) . . . (2n-2i + l) 



1.2 . 

 1 2i.(2i 



1.2 

 =s2j.2« 



Un 



2i 

 Jl(2n 



1.2 



i 



i)..:.(f.+i) \ 



i S 



■2i) 



ri^jp(n(.-i)) 9 Q.-i.Q. 



U2n 



=n(i-i)nt(ft^i)»=f(v«) , » 



We thus obtain the following Table for finding the frequency 

 J n , i of any given form of factor : — 



i = 



h 2, S, 4, 5 



= 1 



1 



= 2 



1 2 



= 3 



9 2 12 



= 4 



225 18 12 144 



= 5 



(105) 2 450 108 140 2880 



The resultant thus determined is the coefficient of the leading 

 term of an equation of the degree l 2 . 3 2 . 5 2 . . . . (2n — l) 2 , upon 

 which depends the determination of a set of 2n quantities f x , 

 f 2 . . . • , f n> so chosen as to make the arc of the curve whose equa- 

 tion is 



y=a,log (*•-{?) +« 2 log (* 2 -?|) . . . +« 2 „log (^-PJ 

 equal to 



a? + a, log 



* + ?i 



+ . . . + a 2 n log 



a? + f s 



* It will, of course, be understood that a x , a 2 , a 3 , &c. are written above 

 in place of a, b, c, &c. 



