OF ARTS AND SCIENCES. 



213 



2. Let cf>he a. real proper orthogonal matrix, then by the theorem 

 above referred to we may put 



<t> 



e\ 



where ^ is a real skew symmetric matrix. 



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

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

 root of 6 having the same multiplicity as II\ since 6 is skew sym- 

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



It may be that among the latent roots of 6 are integer multiples 



of 2 TT V — 1 ; in this case a real skew symmetric matrix 6i can 



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

 root, and such that 



^ 



0^1 



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



± h V- 1 



mo 



± K V- 1, 



m 



"} 



in which mo denotes the multiplicity of the latent root ho = 0, mi de- 

 notes the multiplicity of each of the latent roots ± 7*1 \/ — 1, etc. 

 Let hi, h^, . . . hfj,, be integer multiples of 2 tt, and h^^i, .. . hy, any 

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

 its identical equation is then, if wiq t^ 0, 



F{6) = (6^ + hi^) ($' + Ao^) ....(0' + hj') = O.f 



Let X be any scalar, and let /"I'j x and/"|"' (x) be defined as follows : 



-^ ^ ^ ' \x-h,.\/- y ^x - h, ^-ii x = K V- 1, 



K V- 1, 



* Proc. Lond. Math. Soc, Vol. XXII. p. 453. 

 t Ibid., p. 462. •! 



