﻿out of the Theory of Rectifiable Compound Logarithmic Waves. 381 



of all the resultants so found will be the resultant required, as 

 may be proved by counting its order in the given coefficients, 

 which is easily ascertained to be 



1.3. 5... 2/i + l| 



/l 1 1 \ 



1.3.5.. . Jiw — 1), 



as it ought in order to be the complete resultant. It will be seen 

 then that this complete resultant, like the former one, is still 

 made up of linear factors of the form 



{a 1 + a <2 + ... +a i — a i+l —a i+2 . . . —a. 2i ), 



and it only remains to ascertain the frequency of each such factor. 

 By a calculation precisely similar in nature to that indicated for 

 the case of v=2n, it will be found that for this case of v = 2n -f 1 

 the frequency in question 



= H(i-l)tti . Q(n-i)Q{n-i + l) . 

 For v = 2n it has been already proved to be 



n(i-i)n ? (Q(/z-z)) 2 . 



Thus we obtain the complete double-entry Table of Frequency 

 underwritten : 



i= 1, 2, 3, 4, 5, 6 



= 2 



1 











= 3 



1 











= 4 



1 



2 









= 5 



3 



2 









= 6 



9 



2 



12 







= 7 



45 



6 



12 







= 8 



225 



18 



12 



144 





= 9 



1575 



90 



36 



144 





= 10 



11025 



450 



108 



144 



2880 



= 11 



99225 



3150 



540 



432 



2880 



= 12 



893025 



22050 



2700 



1296 



2880 3628800 



This Table, although obtained by two slightly varying pro- 

 been shown in the text) will easily be seen to be of the form 



#2 — • • • — %'2i — — %2i 



-*2i+2 : 



,= -.r2„ 



X2j+ 1 = 0, dty+a = 0, . . . a? y = 0. 



Hence by a simple enough combinatorial calculation it may be deduced 

 that the number of these fixed possible points of intersection, or, so to say, 



3V+(-l) v 1 



ganglions of the system is 



integer; or, more briefly, the ganglionic exponent is the integer part of ^-. 



8 



\> which is, of course, always an 



