[ 494 ] 



LXXIII. On Pluquatcrnions, and Homoid Products of Sums of 

 n Squares. By the Rev. Thomas P. Kirkman, A.B., Rector 

 of Croft with Southworth, Lancashire*. 

 [Continued from p. 459.] 



THE product Q a Q„,, when simply written out, the sub- 

 stitutions of a single imaginary for certain duads accord- 

 ing to the conditions implied in the triplets being not yet made, 

 is in all cases of the form, 



Q a Q, = (wro,— Safl,) + (2fl * aw i + u,a /) + 2a K H, 



sav 



Q a Q 0/ =(A- B) + (C + D)+ E. 



Let now Q_ a differ from Q„, Q_ fl; from Q a ,,Q_ „ from Q fl/ , 

 only in the signs of all the imaginaries. Then, since a change 

 in the order of the factors Q a and Q„, alters nothing except 

 the sign of E, and a change of sign of all the imaginaries in 

 both the factors alters nothing except the sign of (C-f D), we 

 have 



Q a Q aj =(A-B) + (C + D) + E,- 



Q„, Qa =(A-B) + (C + D)-E, 

 Q_ a Q_ a =(A-B)-(C + D) + E, 



Q_ a ,Q_ = (A-B)-(C + D)-E.. 



(E.) 



Hence, if 



Qa Qa, = Qa„ + R, 



Q_ a; Q_ a = Q_ fl// -R: 



} 



(F.) 



for in Q_ a , Q-aj both c and its equivalent (a b ) have a sign 

 contrary to that which they have in Q a Q„,; and every term 

 in + R appears with a changed sign in the product Q_ a; Q_ a . 



> 



(G.) 



The following are also evident : — 



Q a Q_ a =(A + B) + (C-D)-E,H 

 Q- a/ Qa =(A + B) + (C-D) + E, 

 Q-aQa, =(A + B)-(C-D)-E, 

 Q a/ Q_„ = (A + B)-(C-D) + E. 

 Further, 



QaQ-a=W 2 +a 2 +** +.-.. +y =f* 2 > 



Q*Q-,=w, a + a* +b*+.... +r* =tf, 

 Qa // Q-« / =<-f« // 2 +V + - • • • +V""v/s 



* The indulgent reader is requested to correct the following errors in the 

 preceding part of this paper. 



Page 450, line 14 from top, for e'db read e'bd. 



— 456, — 20 from top, add the terms in. 



— 459, — 4, 14, 18 from bottom, erase all. 



— 459, — 4, 5, 18 from bottom, for fifteen read seven. 



