Sir W. Rowan Hamilton on Quaternions. 29 



If Q and Q' be any two quaternions, the two products of 

 their vectors, taken in opposite orders, namely VQ . VQ' and 

 VQ'. VQ, are conjugate quaternions, by the definition given 

 above, and by the principles of the 9th article ; and the con- 

 jugate of the sum of any number of quaternions is equal to 

 the sum of their conjugates; therefore the products 



(SQ + VQ)(SQ' + VQ') and (SQ'-VQ') (SQ- VQ) 

 are conjugate; therefore T. QQ', which is the tensor of the 

 first, is equal to the square root of their product, that is, of 



(SQ + VQ)(TQ') 2 (SQ-VQ), or of (TQ)*(TQ') 2 ; 

 we have therefore the formula 



T.QQ' = TQ.TQ', 

 which gives also 



U.QQ' = UQ.UQ'; 



that is to say, the tensor of the product of any two quaternions 

 is equal to the product of the tensors, and in like manner the 

 versor of the product is equal to the product of the vei'sors. 

 Both these results may easily be extended to any number of 

 factors, and by using II as the characteristic of a product, we 

 may write, generally, 



TTIQ^nTQ; unQ = nuQ. 



It was indeed shown, so early as in the 3rd article, that the 

 modulus of a product is equal to the product of the moduli ; 

 but the process by which an equivalent result has been here 

 deduced does not essentially depend upon that earlier demon- 

 stration : it has also the advantage of showing that the conti- 

 nued product of any number of quaternion factors is conjugate 

 to the continued product of the respective conjugates of those 

 factors, taken in the opposite order ; so that we may write 



(S-V).QQ'Q"... = ...(SQ"-VQ»)(SQ'~VQ')(SQ-VQ), 



a formula which, when combined with this other, 



(S + V).QQ'Q''... = (SQ + VQ)(SQ' + VQ')(SQ" + VQ")..., 



enables us easily to develope S II Q and V II Q, that is, the 

 scalar and vector of any product of quaternions, in terms of 

 the scalars and vectors of the several factors of that product. 

 For example, if we agree to use, in these calculations, the 

 small Greek letters «, /3, &c, with or without accents, as sym- 

 bols of vectors (with the exception of 7r, and with a few other 

 exceptions, which shall be either expressly mentioned as they 

 occur, or clearly indicated by the context), we may form the 

 following table : — 



