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these remaining terms being easily formed in succession, ac- 

 cording to the lately mentioned law. And to the algebraic 

 sum of all these 210 terms, of which each separately is a po- 

 sitive or negative number, — its positive or negative character 

 depending of course not alone on the prefixed sign + or -, 

 but also on the positive or negative characters of the factors of 

 the product, which enters with that sign prefixed into the 

 term, — Sir W. Rowan Hamilton proposes to give the name 

 of the heterodeuteric, or (more shortly) the Adeuteric 

 Function of the ten vectors a . . a K , for a reason which will 

 presently appear. 



To make the formation of this function of ten vectors more 

 completely clear, it may be observed, that the function of four 

 vectors, which has been above denoted by the symbol 6789, 

 is easily found to represent the sextupled volume of the pyra- 

 mid, whose corners are the terminations of the four vectors (all 

 drawn from one common origin) ; this volume being regarded 

 as positive or negative, according to the character (as right 

 handed or left handed) of a certain rotation; which character or 

 direction is reversed when any two of the four vectors, and, 

 therefore, also, their terminations, are made to change places 

 with each other. On this account the lately mentioned func- 

 tion of four vectors may be called their pyramidal function; 

 and then the foregoing rule for the composition of the adeu- 

 teric function may be expressed in words as follows : — Starting 

 with any one set of four vectors, form their pyramidal func- 

 tion, and multiply it by the aconic function of the remaining 

 six, out of the proposed ten vectors, arranging the vectors of 

 each set in any one selected order. Choose any vector of the 

 four, and any other of the six, and interchange these two vec- 

 tors, without altering the arrangement of the rest, so as to 

 form a new group of four vectors, and another new group of 

 six ; and multiply the pyramidal function of the former group 

 by the aconic function of the latter. Proceeding thus, we 

 can gradually and successively form all the 210 possible groups 



n2 



