48 Sir W. R. Hamilton on some 



then numerous simplifications take place, and the 80 equations 

 between the 24 symbols hnnprstu are found to reduce themselves 

 to 44 equations between the ]5 symbols Imrst, obtained from 

 the five types I. to V. of recent paragraphs, which may be thus 

 denoted and arranged : 



from type I. . . m{ l = lj 3 , r^—r^=s 3 t 1 — s 9 t 3 ; . (166) 



from II. and III. . . m 1 Z ] = m 2 m a , m l r l = m q s l -=m 3 t 1 , . (167) 

 and i,i x /\ 2 = U 2 , m 1 r a =y 8i m l (r- l +r q + r 3 ) = 0; (168) 



from IV. . . s 2 (t- 2 + r 3 ) = - t Y t^ / 3 (r 2 + r 3 ) = - s 3 s v (169) 



and m l s 1 = Ic l t v ?n 1 t l = l 3 s 1 ; (170) 



and from V. . . m 1 s i =l 3 r 9) m 1 t 3 =l q r 3 , li{r- l + r 2 + r 3 )=0. (171) 

 Now these conditions may all be satisfied in each of two prin- 

 cipal ways, conducting to two distinct systems of associative 

 quines, which may be called Systems (A) and (B), but which are 

 not the only possible systems of such quines, because we need not 

 have commenced by assuming the equations (165), although that 

 assumption has simplified the problem. For first we may sup- 

 pose that the constants / and m are different from zero, but that 

 the constants r are connected by the relation 



(A).. r l + r q + r 3 =0; (172) 



or secondly, we may reject this relation between the constants r, 

 and suppose instead that the six constants I and m all vanish, so 

 that 



(B).. /, = / 2 = / 3 =7w 1 = ??? 2 =m3=0. . . (173) 



With the first supposition, (172), we are to combine the nine 

 relations between the fifteen constants Imrst, which are suffi- 

 ciently expressed by the formula (167), or by the following: 

 (A x ) . . l i = m~ 1 m 2 m 3 , s l = m~ 1 m 1 r l , t x = m~ 1 m ] r 1 ; (174) 



and then all the other conditions of association will be found to 

 be satisfied, if we equate each of the ten symbols abc to zero, or 

 if we establish this other formula, 



(A 2 ).. 0, = O, b, = 0, c,=0, ff 4 =0: . . . (175) 

 while there will still remain five arbitrary constants of the system, 

 for instance i\r q m x m q m 3 . With the second supposition, (173), 

 we are to combine four distinct relations between the nine con- 

 stants rst, contained in the formula (169), or in the following: 



(B,)*.. ?-, + r 2 =- s rV3> «iV3=W 3 ; • • (176) 



* It must be observed tbat tbese equations (17^), which are part of the 

 basis of the system (B), are true in the system (A) also, as corollaries from 

 (17-1) and (172), which last equation does not hold in (B) ; and which 

 allows us to reduce, for (A) but not for (B), the relation (177) to the simpler 

 form rir 2 r 3 =sis 2 s 3 . 



