143 



The following rule is suggested to determine, in any complicated 

 case, whether the number of contiguous interchanges by which one 

 arrangement of letters is converted into another shall be odd or even. 

 This is an important matter in the theory of eliminants, though very 

 complicated instances may seldom occur in practice. Write down 

 one arrangement under the other, and, beginning at one letter in one 

 line, mark the companion letter in the other line, pass on to that 

 companion in the first line, mark its companion, and so on, until we 

 arrive at a letter already marked. Call this sequence a chain, each 

 mark being one link. Having formed one chain, begin at a letter 

 not yet used, and form another ; and so on until every letter has 

 been used. Then, according as the number of chains with even 

 links is odd or even, the number of interchanges of contiguous letters 

 required is odd or even. For example, the two arrangements being 



ABCDEFGHIJKLMNOPQ 



HMOGQBKLJPFC I NADE 



121 2 3 221222124 12 3. 



Under A is H, under H is L, under L is C, under C is O, under 

 O is A, already taken : the first chain has five links, the second is 

 found to have nine, the third two, the fourth one. The number 

 having even links is one, an odd number ; hence an odd number of 

 contiguous interchanges converts the first arrangement into the 

 second. 



23. The following is the method of ascertaining whether the bior- 

 dinal equation 



Q + Rr + Ss+Tf+U(s»-rO=0 .... (1) 



possesses a primordinal of the form f(x, y, z, p, q) = 0, containing 

 an arbitrary function. Considering x, y, z, p, q as five independent 

 variables, integrate, by common methods, the equations 



u(*+.*Ut- — ^-S^=0 



\dx dz) dp l + k dq 



u^ +? ^ + R---^-S-=o, 



\dy dz ) dq \-\-k dp 



k being one of the roots of &S 2 =(1 + A) 2 (RT4- QUJ. If a common 

 solution u = A can be found, then A=const. is a primordinal of (1). 

 If two common solutions, A and B, can be found, then B=zjA is 

 a primordinal, ot being arbitrary. But though in this case A=const. 

 and B = const, are solutions, they cannot coexist, unless the values of 

 k be equal, or unless S 2 =4(RT+QU). This last equation is one 

 condition of polarity ; and if, when satisfied, we find three (and there 

 cannot be more) common solutions, A, B, C, inexpressible in terms 

 of each other, then /(A, B, C) = is the most general primordinal, 

 any two forms of it may coexist, or even any three, which amount 

 to A = const., B=const., C = const. Elimination of p and q between 

 these last equations gives (p(x, y, z, a, b, c)=0, the modular equa- 



