300 Rev. T. P. Kirkman on Bisignal Univalent Imaginaries, 



of the required form Q" 23 ; from which it follows, that the pro- 

 duct of two sums each of twenty-four squares can be reduced 

 algebraically to a sum of twenty-four squares, if fifteen con- 

 ditions are satisfied ; or that 



(w 2 + a 2 + b 2 +..+g 2 + h 2 + h /2 + i 2 4-i /2 + .-+p 2 H-p /2 )(w / 2 + a / 2 



-fb / 2 + ..-}-g / 2 + h y 2 + h/ 2 + i / 2 -fi/ 2 + ...-f-p / 2 + p/ 2 ) 

 is a sum of twenty-four squares, if 



h : h i= =h' : h/=i : i y =i' : i',= ... =p : P/ =p' : p/. 



We can readily prove, by adding continually a new group 

 of twenty-eight triplets made with eight new imaginaries, giving 

 rise to twenty-eight new bisignal duads, that the product of 

 two sums each of 8{n-\- 1) (?i > 0) algebraic squares can be re- 

 duced to a sum of 8(rc+l) algebraic squares, if the given 

 squares satisfy 8n — 1 conditions. Hence we have the follow- 

 ing theorem : 



The product of (r + 1 ) sums, each of 8 (n -f 1 ) algebraic squares 

 (n > 0), can be reduced to a sum of8(n+l) algebraic squares if 

 the given roots satisfy 8nr—r assignable simple equations. 



In the case of 2 n — 1 imaginaries, triplets can be formed by 

 which every duad may have an equivalent monad, and this 

 consistently with a law that must pervade every congruous 

 system of such triplets : namely, this law, that if abc and ahi 

 be two triplets, giving values a = bc—hi, both bi=ch and 

 bh = ic must hold good; that is, bi and ch must be completed 

 into triplets by the same imaginary m, and bh and ic must be 

 combined with the same monad n. 



Such a system is completed for thirty-one imaginaries by 

 adding to the groups ABB' the following, the signs being 

 here of no importance. 



faqr bqs cqt dqu eqv fqw gqx 



J ast brt crs drv eru frx grtso 



| auv buw cux dsw esx fsu gsv 



\.awx bvx cvw dtx elw ftv gtu 



c<! 



bqs 



cqt 



brt 



crs 



buw 



cux 



bvx 



cvw 



ih'r 



kh's 



ii'q 



ki't 



ik't 



M'q 



il's 



Mr 



hh'q ih'r kh's Ih't mh!u nh ! v oh'w ph'x 



hi'r ii'q ki't li's mi'v ni'u oi'x pi'w 



h¥s ih't M'q Ih'r mk'w nli'x oh'u pk'v 



hl't il's kl'r IV q ml'x nl'w ol'v pl'u 



hm'u im'v km'w Im'x mm'q nm'r om'r pm't 



hn'v in'u hn'x In'w mn'r nn'q on'q pn's 



ho'w io'x ko'u lo'v mo's no't oo't po f r 



Jip'x ip'w hp'v Ip'u mp't np's op's pp'q* 



