RoGEKS — Mode of Representing Linear Orthogonal Transformation. 69 



We must show now that when n is even /(A) = is a reciprocal equation, 

 and therefore may be reduced to one of degree —. ■ 



19. Independent i'avariants of a motion. — (1) Let n be even, and suppose 



/(A) = A" - AiA"-i + A.A''-^ - . . . + A„_^\- - A„_jA + 1. 

 The absolute term = A = 1. Xow A,- is the swn of leading minors of order 

 r of A. But since A = 1, each leading minor is equal to its complementary 

 minor, since A is identical with its reciprocal. Hence A,- = A,,.,- There are 



thus "5 independent invariants Aj, Ag. A3, • . • A«, ivhen n is even* and /(A) 



may be written in the reciprocal form 



» 



A" + 1 - Ai (A"-i + A) + . . . ± A'^ A„. 



It 



Dividing by X^ and putting X + t = 2 cos 6, we reach the following 



A 



equation, which may be expressed as an equation of degree 7; in cos 6 : — 



2'" cos m% - 2"'-iAi cos {m - 1) 6 + . . . + A„, 

 where 1m = n. 



(2) Let n be odd and = 2m + 1. Then in like manner A,- = A,,^,-, and 

 there are ni independent invariants (equal in number to that of the angles) 

 Aj, Aj, . . . Am.f There is no central term, hence 



f(X) = -X"+l + Ai (A"-i - A) + . . . + A„, (A"'^i - A'"-'). 



Hence /(I) = 0, and we may divide across by A - 1. By putting 



A + r = 2 cos 0, 

 we get an equation of the wi** degree for cos H. 



20. Determination of the rotatory flats. — Suppose first that n is even, and 

 that we refer to the self-returning origin (§ 21). This amounts to finding an 

 orthogonal substitution of the form 



Jl = a^x + j3i2/ + yis + Si« + . . . . 



Y = n^ -h- lizy + fiZ + SjZt + . . . . 

 Z = a^ + . . . . 



W= ttiX + . . . . 

 etc., 



* That they are independent in general may be proved by noting that the canonical 

 form is possible with arbitrary values of S, (p, . . . and that there must be the same 

 number of independent invariants as there are angles. 



t There is also an invariant of translation (§ 23). 



