48 Proceedings of the Royal Irish Acadmyiy. 



The function C may be called a combinatorial function, for 



G{p + xq,q) = Q{pq\ G{pp) = 0. (6) 



' Thus an arbitrary bilinear function is reducible in one way to the 

 sum of a permutable and a combinatorial function. 



4. For all quaternions ji?, q, r, we agree to write 



M{qr) = ^qfXpr) = M"{qp); (7) 



and we call the new functions f\pq), f"{pq), the Jirst and second 

 conjugates, respectively, of the given function. The phraseology is 

 justified by the consideration that the first conjugate is the conjugate 

 if the bilinear function is regarded as a linear function of its first 

 quaternion. 



5. Permuting the quaternions in (7) according to the rule (3), and 

 taking the conjugates, we obtain the series of equal scalars 



SMf) = ^qf'ipr) - ^r{f')"{pq) = iiq{f),{rp) 

 = M"{qp) = ^r{f"), ipq) = ^q{f")\rp) 

 = ^pfM) -^r{l)'{pq) =^q{f,)"{rp); (8) 



and from these we obtain the relations 



(/T(m) = Lf"\{pq) = LQ'{pq) = /"{qp), 



(A ipq) = LfJipq) = {l)"{pq) = f'iqp) ; (9) 



where the brackets are employed to obviate any confusion as to the 

 order in which the operations indicated by the accents have been 

 performed. 



These relations, taken in conjunction with the obvious relations 



f(P9) = LQ.ipq) = {fJipq) =- ifTipq), (lo) 



enable us to reduce any multiply-accented function to one or other of 

 tlie six fundamental functions, — the function and its two conjugates, 

 the permutate and its two conjugates. 



6. Having now explained the fundamental principles underlying 

 the manipulation of bilinear functions, we shall indicate some of the 

 uses to which they may be applied. 



