Sir W. Rowan Hamilton on Quaternions. 291 



which seems to him to open a wide and new field of analytical 

 research, connected with many important and difficult de- 

 partments of the mathematical study of nature. 



A QUATERNION, Symbolically considered^ being (according 

 to the views originally proposed by the author in 184-3) an 

 algebraical quadrinomial of the form iso + ix+Jy + kz, where 

 wxi/z are any four real numbers (positive or negative or zero), 

 while ijk are three co-ordinate imaginary units, subject to the 

 fundamental laws of combination (see Phil. Mag. for July 

 1844-): 



ij=k;Jk:=i; /ci=J; I . . . . (a.) 



ji^—k; kj=—i\ ik——j;J 



it results at once from these definitions, or laws of symbolic 

 combination, (a.), that if we introduce a new characteristic of 

 operation, < , defined with relation to these three symbols ijk, 

 and to the known operation of partial differentiation, performed 

 with respect to three independent but real variables xyz, as 

 follows : 



id ;d ^d ,, . 



'^ = T.+% + T.' (''•' 



this new characteristic < mil have the negative of its symbolic 

 square expressed by the following formula : 



of which it is clear that the applications to analytical physics 

 must be extensive in a high degree. In the paper* designed 

 for Southampton it was remarked, as an illustration, that this 

 result enables us to put the known thermological equation, 



d^t> d?v d^u dv 



db^+d^ + d^ + ^'dF"^' 

 under the new and more symbolic form, 



(<=-r')-'" (^0 



while < V denotes, in quantity and in direction, the flux of 

 heat, at the time t and at the point xy%. 



50. In the Proceedings of the Royal Irish Academy for 

 July 1846, it will be found to have been noticed that the same 

 new characteristic < gives also this other general transforma- 

 tion, perhaps not less remarkable, nor having less extensive 



* In that paper itself, the characteristic was written V ; bi't this more 

 common sign has been so often used with other meanings, that it seems desi- 

 rable to abstain from appropriating it to the new signification here proposed, 



U2 



