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V. 



THE MULTI-LmEAR QUATERNIOX FUNCTI0:N\ 



By CHARLES JASPER JOLY, M.A., D.Sc, F.T.C.D., Royal 

 Astronomer of Ireland, and Andrews' Professor of Astronomy in 

 the University of Dublin. 



[Read NoTEMBEK 10, 1902.] 



1. A bilinear quaternion function is symbolically defined by the 

 equation 



f{a + h, e + d) =f{a, c) +f{a, d) +f{b, c) ^f{b, d), (1) 



a, b, c, and d being any four quaternions. In other words, the function 

 f{Pi 9) is distributive with respect to its first quaternion j;?, and also 

 with respect to its second quaternion q. 



The quaternions e being arbitrarily assumed constants, the function 

 may be expressed in the form 



f{pq) = eSpM + eSpfii + eSpM + ^SpM ; (2) 



and is thus seen to involve sixty -four constants, sixteen in each linear 

 function /i, f^, /s, f^. 



2. Transposing the quaternions alters the function into its per- 

 mufate, which may be distinguished by a sub-accent ; thus 



f{Pi)=fMp\ f{qp)=f.{pq)' (3) 



If the linear functions in (2) are self-conjugate, the function is 

 permutahle, and conversely. 



3. Introducing two new functions P and C defined by the equa- 

 tions 



npq) = :2{pq) + C{pq), Mpq) = ^ (pq) - C{pq), (4) 



it is evident that P is a permutahle function, and that C changes sign 

 with permutation; in fact (by (3) and (4)), 



2P ipq) =f{pq) +J\qp\ 2C {pq) =f{pq) -f{qp)- {O) 



K.I. A. PKOC, VOL. Vm., SEC. A.] F 



