Mr. John T. Graves on the Theory of Couples. 31 7 



If we assume 



(7.) {i X +jy) {i oTi +j>i) = i x^ +jy^, 



we shall obtain from (3.) and the equations (A.),' 

 (8 ) •^•2=«-*'^'i+'*('^^i+i/*^i) + ^'3/^i\. 



In normal couples, the equations 



(9.) i^ .j = i . ij, / 'i-j' ij, 



hold good by virtue merely of the assumed rules of multiplica- 

 tion^ without the necessity of assuming any linear relation 

 between i and j. We proceed to investigate the relations 

 which must subsist among the constants a, a', /3, j3', x, x', in 

 order to satisfy this condition. 



Since t^j = i . ij =j . i^, we have, by (5.) and (4.), 



(10.) i^x + ijxJ = ijx +j^^. 



Since J^ 2 =^j 'ij =■ i'j% we have, by (5.) and (6.), 

 (11.) ijx +/ x' = i^ /3' -\- ij a'. 



Substituting in (10.) and (11.) the values of 2*^, ij and 7'^ 

 given by equations (A.), we get from (10.), 



(12.) {iu +j^) X + {i x+jx') x' = (z X +Jk') « + (/jS' +ja!) /3. 

 Similarly, we get from (11.), 



(13.) {ix+jx') X + {i^' +ja}) x' = {icc+j^) /3' + {ix+jx') a!. 

 Arranging (12.) and (13.), we obtain 



(14.) ?(xx'-/3/3') +7(x'(x'-«)-/3(«'-x)) =0, 

 (15.) j(xx'-/3/3') + /(|3'(x'^a) -x(«'-x)) = 0. 



It is evident that the two equations (14.) and (15.) will be sa- 

 tisfied, independently of any linear relation between z andj, if 

 we suppose, in conformity with the characteristic rule of 

 couples, 



(16.) xx' — (3/3' = x'(x' — a)~^(a'— x)=/3'(x'— a)-'<(a'-'t) = 0. 

 The three equations (16.) are (as it is not difficult to per- 

 ceive) equivalent to the two following : 



(17.) ^ -x'-a- x' 



The two equations (17.) express the relations which, in 

 every system of normal couples, connect the constants of mul- 

 tiplication. It is easy to see that the equations (17.) will be 

 satisfied by integer values of the constants, if we suppose that 



the three equal fractions in (17.) are each equal to — — ; and 



if we take 



(18.) K'=pgi^ = -g§,ct'-x-pri,x'-a=-qri,^'--pQ,x = qQ. 



