190 



Sir W. R. Hamiltois^ LL. D., read the foUowing paper : — 



On the Existence of a Symbolic and Biquadeatic Equation, which 

 IS satisfied by the Symbol of Lineae Opeeation in Quateenions. 



1. In a recent communication (of June 9, 1862), I showed how the 

 general Linear and Q-naternion Eunction of a Quaternion could be ex- 

 pressed, under a standard quadrinoial form ; and how that function, 

 when so expressed, could be inverted. 



2. I have since perceived, that whatever form be adopted, to repre- 

 sent the Linear Symhol of Quaternion Operation thus referred to, that 

 symhol always satisfies a certain Biquadratic Equation, with Scalar Co- 

 efficients, of which the values depend upon the particular constants of the 

 Function above referred to. 



3. This result, with the properties of the Auxiliary Linear snad Qua- 

 ternion Functions with which it is connected, appears to me to consti- 

 tute the most remarkable accession to the Theory of Quaternions proper, 



. as distinguished from their separation into scalar and vector parts, and 

 from their application to Geometry and Physics, which has been made 

 since I had first the honour of addressing the Royal Irish Academy on 

 the subject, in the year 1843. 



4. The following is an outline of one of the proofs of the existence 

 of the biquadratic equation, above referred to. Let 



fl^r (1) 



be a given linear equation in quaternions ; r being a given quaternion, 

 q a sought one, and / the symbol of a linear or distributive operation : 

 so that 



.fiS + l')-fl+fs'. (2) 



whatever two quaternions may be denoted by q and q\ 



5. I have found that the formula of solution of this equation (1), or 

 the formula of inversion of the function, f may be thus stated : 



nq = nf~^r = Fr ; (3) 



where nis a scalar constant depending for its value, and i^is an auxili- 

 ary and linear symbol of operation depending for its form (or rather for 

 the constants which it involves), on the particular form of /; or on the 

 special values of the constants, which enter into the composition of the 

 particular function, fq. 



6. We have thus, independently of the particular quaternions, q and 

 r, the equations, 



Ffq = nq, fFr = nr ; (4) 

 or, briefly and symbolically, 



Ff=fF^n. (5) 



7. Changing next / to /, = / + that is to' say, proposing next to 

 resolve the new linear equation, 



