70 Proceedings of the Royal Irish Academy. 



such that the original equations of motion 



ai. = hx + mi?/ + etc., 



take the canonical form (§ 17). N"ow this would give rise to n- linear 

 equations, which it would be very difficult to solve and harmonize even if 

 n = 4. The equations will be n of the form 



litti + 4^1 + /371 + .... = Ui cos B + Oi sin Q, 



where on the left I is replaced by m, n, p, etc., in turn, and on the right « is 



replaced by /3, 7, 5, etc., in turn ; n more equations similarly constructed of 



the form 



A02 + h^2 + ^72 + .... = - a, sin 6 + Or; COS a. 



The remaining equations will similarly involve tlie other angles (^, 1^, etc.) 

 in groups of 2m for each angle. We shall also require 



2a/ = 1, 5f(,a3 = 0, for all values of r and s (?• ={= s). 



The use of the determinaut/(A) simplifies the work in a truly astonishing 

 manner. Let nx + ifX2 be a root Ai of /(X) = 0, corresponding to 6, where 

 i- = - 1. Let 5i, ))i, ^1, wi, . . . be the corresponding ratios which satisfy 



h^i + rnvh + n\ ?i +lh'^i +.... = ((Ui + i/Uj)?] 



4?i + tihriT, + = {lJ-i + -i/ti:)))! 



h%i + = (i"i + ^^"2)^1 



Summing the squares of both sides 



?,- + Iff + ^r + .... = (/i, + ■i//2)= (?r + »/i' + ?,= + ... .). 

 But the transformation is orthogonal, hence 



SV = Ar2?/^ 

 and since Ai generally is not = 1, we have* 



n:- = 0. 



Let the values of ?,, ^^, ^, . . . , , and their conjugates Ij, ))2, ?2 . . . . , found 

 from the linear equations be 



£1 = tt) + ia-!, ^2 = "1 - io-i 



j)i = ^i + i/Sz ih = /3i - 1/32 



S: = 7i + *72 Sj = 71 - ■iyj 



etc. etc. 



Hence, since 2?,- = 0, 



SaiQo = 0, 5*01^ = "^u-i = 1, 

 for we may take 1 as their common value. 



* If one or more roots = 1, we have a singularity not here considered (see Note, p. 73). 



