Sir W. Rowan Hamilton on Quaternions. 27 



simply the vector of the quaternion ; and may be denoted by 

 prefixing the characteristic Vect., or V. We may therefore 

 say that a quaternion is in general the sum of its own scalar 

 and vector parts, and may write 



Q = Seal. Q + Vect. Q = S. Q + V. Q, 

 or simply 



Q = SQ + VQ. 

 By detaching the characteristics of operation from the signs 

 of the operands, we may establish, for this notation, the ge- 

 neral formulae : 



1=S + V; 1-S=V; 1_V = S; 

 S.S = S; S.V = 0; V.S = 0; V.V = V; 



and may write 



(S + V)" m l, 

 if n be any positive whole number. The scalar or vector of 

 a sum or difference of quaternions is the sum or difference of 

 the scalars or vectors of those quaternions, which we may ex- 

 press by writing the formulas : 



SS = 2S; SA = AS; V2 = SV; VA = AV. 



19. Another general decomposition of a quaternion, into 

 factors instead of summands, may be obtained in the follow- 

 ing way: — Since the square of a scalar is always positive, 

 while the square of a vector is always negative, the algebraical 

 excess of the former over the latter square is always a posi- 

 tive number ; if then we make 



(TQ) 2 = (SQ) 2 -(VQ) 2 , 

 and if we suppose T Q to be always a real and positive or 

 absolute number, which we may call the tensor of the quater- 

 nion Q, we shall not thereby diminish the generality of that 

 quaternion. This tensor is what was called in former articles 

 the modulus * ; but there seem to be some conveniences in not 



* The writer believes that what originally led him to use the terms 

 "modulus" and "amplitude," was a recollection of M. Cauchy's nomen- 

 clature respecting the usual imaginaries of algebra. It was the use made 

 by his friend, John T. Graves, Esq., of the word " constituents," incon- 

 nexion with the ordinary imaginary expressions of the form x-\- V — ly, 

 which led Sir William Hamilton to employ the same term in connexion 

 with his own imaginaries. And if he had not come to prefer to the word 

 " modulus," in this theory, the name " tensor," which suggested the cha- 

 racteristic T, he would have borrowed the symbol M, with the same signi- 

 fication, from the valuable paper by Mr. Cayley, " On Certain Results re- 

 specting Quaternions," which appeared in the Number of this Magazine 

 for February 1845. It will be proposed by the present writer, in a future 

 article, to call the logarithmic modulus the "mensor" of a quaternion, and 

 to denote it by the foregoing characteristic M; so as to have 

 MQ = log.TQ, M.QQ'-MQ + MQ'. 



