OP ARTS AND SCIENCES. 



213 



2. Let c/> be a real proper orthogonal matrix, then by the theorem 

 above referred to we may put 



</> 



e°, 



where d is a real skew symmetric matrix. 



Since 6 is skew symmetric, its latent roots occur in pairs opposite 

 in sign ; that is, if H is a latent root of 6, then — H is also a latent 

 root of 6 having the same multiplicity as H; since 6 is skew sym- 

 metric and real, its latent roots are purely imaginary.* 



It may be that among the latent roots of are integer multiples 

 of 2 7r V — 1 ; in this case a real skew symmetric matrix $i can 



always be found of which no integer multiple of 2 tt \ / — 1 is a latent 

 root, and such that 



4> = e e = A 



Thus, let the latent roots of 6 be given by the following schedule : 



Latent root, 

 Multiplicity, 



Latent root, 

 Multiplicity, 







m. 



±^iV-i 



i>h 



±hV-i 



m.> 



± K V- 1 



m„ 



±/v+iV-i 



m 



M + l 



± K V- 1, 



//( 



v> 



in which ?n denotes the multiplicity of the latent root h = 0, m x de- 

 notes the multiplicity of each of the latent roots ± h x V — 1, etc. 

 Let h u h 2 , . . • h,j., be integer multiples of 2 it, and h li + 1 , . . . h v , any 

 real quantities other than integer multiples of 2 tt. Since 6 is real, 

 its identical ecpuatiou is then, if m 4 0, 



F(6) = (0* + V) (0 2 + K) • • • • (# 2 + V) = O.f 



Let x be any scalar, and \et f^x andy*|. 2) (x) be defined as follows: 



Jr k ) \ x _ K ^zrO \ x -h r ^-i' x = h r v-h 

 Jr() \ x + h r V-i J \x + h r V-i J x = -KV-i, 



* Proc. Lond. Math. Soc, Vol. XXII. p. 153. 

 t Ibid., p. 462. 



