(139) 



266 Sir W. R. Hamilton on some 



[23] =231 . 122 + 233 . ^22^l^n^-n^m^', 



[231] =211. 311+231. 313= --?W2"3 + A^u 

 [32]=321.133 + 322.233 = /,m,-?n3W2; 



[32]]=311. 211 + 321. 212=-WgW2 + /i/i,;. 



but, whether we equate the first to the fourth, or the second to 

 the third of these expressions for Z>j, in conformity with the type 

 (134), we obtain only one common equation of condition, WjWig 

 = W37W2, equivalent indeed by cyclical permutation to three, 

 namely to the following, 



= WgWg — n^m^ = WgWi, — n^m^ = n^m^ — n^^ ; . (140) 



which evidently agree with certain simple combinations of the 

 six equations on the two last Hnes of (86) . If however we com- 

 pare either the first value (139) with the third, or the second of 

 those values with the fourth, according to the type (137) or (138), 

 we find by each comparison the common condition l^n-^=^l^m^, 

 and thus recover the three equations of the first line of (85). In 

 this way then we may obtain the required number of six distinct 

 equations, with two terms each, between the nine symbols [fgh), 

 or /, . . . Tig, for the case of quadrinomes, by elimination of the 

 three symbols [fg), or of ^j, Z>2, b^. 



[21.] For the case of quines (w=4), the general theory requires 

 that the corresponding elimination of the 6 -^n[n — \) symbols 

 of this form [fg), or b^ ., » Cg, should conduct to 24=n(7i~l) 

 (w — 2) distinct equations of condition, with 4=2(w — 2) terms 

 each, between the ^n^(/i~l) = 24 symbols of the form (fgh), or 

 /, . . . Wg, each equation thus obtained being homogeneous, and 

 of the second dimension ; and that all these 24 conditions should 

 be included in the formula (134), or in the single type (123). 

 And in fact we thus obtain, by comparison of the six expressions 

 for ij, of which one is 



Z,, = (23) = [23]=2;(23A. A22) = /i7ii-W2mg-j3ir2, (141) 



the four following equations of condition, included in that type 



or formula : 



0= [23] - [321] = [32] - [231] H 



0= [23] -[324] = [32] -[234]; J * * ^ ^ 



that is, with the notations /j . . t/g, 



n^m^ + WgWg =J03*3 + llm^ = l^n^ —p4<i i -" 



while we have in like manner six expressions for Cj, of which 

 one is 



c,= [41]=S(4U . M4)= -(r,M,+5iW2 + /,W3), . (144) 



