The Rev. T. P. Kirkman on Pluquaternions. 495 



for every term of the product Q„ Q_ a which ia not in /x 2 , is 

 cancelled by virtue of the property common to all the 2n+l 

 imaginaries, 



m o n o + n o m o ^0. 

 Wherefore 



Now since Q a/ Q- fl/ is real, Q-a'Q^Q-a/ cannot differ from 



Qa/.Q-a/Q-a; for 



A + \/^lB = A + B */~\ ; 



and A-f V — IB is a case of Q_ a ' Qa/Q- a/ > namely that case 

 in which the (2rc-f 1) imaginaries are reduced to any one of 

 them. We say boldly, then, 



whence 



^ 2 =Q fl Q-»Q«,Q-«,=Q a Q« ( Q-,Q-a=(Q» i( + R)(Q_ fl/ -R) 

 = Q*„Q- a , - Q a/ R + RQ_ 0/ - RR. 



— Qa„R -f RQ_ fl// represents a sum of imaginary terms of the 

 form — (c Q 'b n o + b ff ?&)£*@*» That these are none of them 

 real, is plain from the consideration that b n has no equiva- 

 lent single imaginary, whence that b « = +c Q is impossible. 



— RR has two kinds of terms. One kind is a sum of ima- 

 ginary quantities of the form — (b a n a ' d l + d l 'b n )B n D„ 

 which cannot be real; for if b n 'd l had any real value, 

 b n would have a value different from that equivalent duad 

 which alone it can represent, and which does not appear in 

 the condition-function R. The other kind are all of the form 

 —b i 'b i B 4 a . 



Now 



h *o ' ^o *o = — *o ^o* ^o *o = — *V &o* &o' *o> 



= - V bo* 'io=— V io K*> 

 since bj 1 is real ; 



Let 



B*=B. 2 +... + B, 2 + + B* + &c, 



or = the sum of the squares of all the real quantities in the 

 condition-function R : then, equating real quantities, we have 

 the very interesting result, 



or the following 



