Mr. John T. Graves oti the Theory of Couples. 319 



equations (B.). Eliminating ^ between (20.) and (21.), we 

 get 



(29.) 7 = '^-'(P^ + ^'i) = JXL. 



Hence we find upon arranging, 



(30.) i^-i^ p q + qri) i+ {p^ q^p qri + q^ ^) g = 0. 



In like manner, by eliminating / between (21.) and (22.), we 

 find 



(31.) f-{^q^+pri)j+{p^g+pgyi + g''^)& = 0. 

 Upon solving (30.), we get 



(32.) i = ^PQ + q^tQ^f-"^?^ ; 



and similarly, upon solving (31.), we get 



\ 'J J 2 



It will be found upon examination, that, in order to satisfy 

 the equation (21.), we must make the positive sign of the radical 

 in (32.) correspond to the negative sign in (33.), and vice versa. 



Let the two values of i given by (32.) be denoted by /, and 

 ?2, and let the corresponding values of 7 be denoted hyj\ and 

 72. Let 



(34.) p^g+pqrj + q'^Q = u, 



and let 



Then we shall have (as will be found upon trial) the three 

 following connected equations : 



(36.) ii ii — u gl 



(37.) /ii2+ii^2 = «'3 ^ (E.) 



(38.) i,i2 = «sj 



Hence we obtain the leading theorem, 



(39.) {hp^ +ii 17J [HVn +.h 9n) = '^ V 

 Further, we have, by the equations (C), since both i^yj\ and 

 hiJi satisfy the equations (B.), 



(40.) (i, p, +j\ qi) (/, p^ +j\ q^) = / 1 ^3 4-ii qs~\ ^ 



(41 .) [t^p^ +j^ q,) {i^pc, +^2 9<2) = hP3 +J^ 9s J ' 



Pq and g'g having the same values as in (24.) and (25.), or, as 



in (27.) and (28.) 



Multiplying together (40.) and (41.), we get by (39.), 

 (42 .) M Wj . M Mg = u f/g ; 



and, dividing both sides of (42.) by u, we obtain, finally, the 



interesting result. 



