318 Mr. John T. Graves on the Theory of Couples. 



By these assumptions we are enabled to make the six con- 

 nected constants, «, «', |3, j3', x, x', depend upon five arbitrary 

 and unconnected constants, p, q, g, >j, 9. Certain limited va- 

 riations of sign might be allowed in our last assumptions con- 

 sistently with the integer character of the constants of multi- 

 plication. Thus it would appear more natural to assume the 



equal fractions in (l7.) each equal to + — instead of — — , 



... 5' y 



and to give positive values to every numerator and denomi- 

 nator, taking 



(19.) x'=:p§i ^—qqi a.' — )i=:pYif x'—a—qviy ^'=p^, x — q^. 



To this there would be no objection on principle, and the 

 results to which it would lead would be convertible with those 

 which follow. The assumptions (19.) were, in fact, nearly 

 those which I at first made, but I found upon trial that the 

 signs of (18.) are those which give the greatest concinnity to 

 the subsequent calculations. 



By substituting in equations (A.), in place of the connected 

 constants of multiplication, their values as determined by 

 equations (18.), we get 



(20.) i^ = i{pg+qri)-Jqg-\ 



(21.) iJ^iqQ+Jpg L . . . (B.) 



(22.) f=j{qQ + pyi)-ipU 



From equations (B.) we get the following system of equa- 

 tions, defining a general rule of multiplication to which all 

 normal couples will be found to conform : 



(23.) {ipi +Jqi) {iPi+Jqc,) = iPa+JQs 1 



(24.) P3=^(jPg + qri).PiPc,+ q9,{Piqi + qiP^)-pLq,qc^ k (C.) 

 (25.) 5'3= {qS+p ))) . q, qc,+p § (Piq^ + qiV^) -q§-PiPj 

 If we express the couple ip+jq by the notation (p^ q), and 

 arrange equations (24.) and (25.), the multiplication of normal 

 couples will be expressed by the following system : 



(26.) (p„ q,) (^2, ^2) = iPs^ Q3) '] 



(27.) P3=ppiP^g+gPiP2^ + {(iPi92+gQiP9-pqi9i)^ ^ (D.) 

 (28.) 5'3=?2'i5'2^ +pgiq^^+{pqiP2+pPiqi'-9P^Pd?^ 



A connection between the theory of multiplication of nor- 

 mal couples and the resolution into factors of quadratic func- 

 tions of two variables, will appear from the investigation to 

 which I now proceed. Thus the theory of couples becomes 

 connected with that of quadratic equations. We may treat i 

 and J as ordinary algebraic quantities, and endeavour to de- 

 duce their values in terms of p, q, g, )j, 6, by means of the 



