316 Mr. John T. Graves on the Theory of Couples, 



original paper on Tri-plets (published in the Cambridge Phi- 

 losophical Transactions), have induced me to recur to the ex- 

 amination of Couples. In the present paper I shall confine 

 myself to the algebraical consideration of the subject. 



Couples are of the form ix-[-jy; the variables x and y are 

 called the constituetiis of the couple, the constants i and j its 

 characteristic coefficients. The couples that I treat of are sup- 

 posed to follow the analogy of ordinary algebra, subject to the 

 characteristic rule, that " if two couples are equal, their con- 

 stituents are separately equal." We are precluded from sup- 

 posing the existence of any linear relation between the coeffi- 

 cients. If we have the equation 



(1.) ix+jy:=ix^+jy^, 



we are to assume ^ s= j^j, j/ = ^j, and are not allowed to esta- 

 blish the equation / =7, "^^^ — ~, '\^ — — - have a determinate 



value. Hence, if we have ix +jy = 0, we are bound to as- 

 sume .r=:0, j/ = 0. This view of couples is derived from the 

 ordinary algebra of imaginary quantities. If 



(2.) X + V'^ly^x^Jr V'^y^, 



we have x ■= x^ and y = y^ 



We are to assume such rules that functions of couples may 

 be couples. The product of two couples is a couple. If we 

 follow the rules of algebra, we shall have 



(3.) {ix +jy) (z>, +7>i) = i^\vx^ + ij {xy^ +3/^^ +fyy^. 



Sir W. Hamilton, in quaternions, by a happy deviation 

 from the rules of ordinary algebra, makes ij=- —Jit but we are 

 at present treating of couples which obey ordinary algebraic 

 rules. Possibly science may gain more by the introduction 

 of ajiomalous couples, but I confine myself at present to normal 

 couples. 



In order that i^xx^ + ij{xy^ +.^-^'i) '^J^yVv* which is the 

 product of two couples, may be itself a couple, we must define 

 2^, ij and J^ in such a way as to reduce the product to the 

 form ix^-\-jyc^. This cannot be effected unless we suppose 



(5.) z7=/x+7x'L (A.) 



(6.) / = //3'+i«'J 



a, ^a, /3, '/3, x, 'x being constants capable of being connected 

 by linear relations. They may be termed •■' the connected 

 constants of multiplication." 



