334 On the Contacts of Lines and Surfaces of the Second Ordei\ 



body, another step in the hierarchy of aggregation is to be taken 

 into account, and that we must know for this purpose not only 

 the number of molecules in the body and the number of atoms 

 in each molecule, but also the number of monads in each atom ; 

 the number of bodies (differing by definition) capable of being 

 formed out of n monads will then represent what I mean by the 

 index of double decomposition of (or if we like so to say) to the 

 modulus n. And it is obvious that this idea admits of indefinite 

 extension, and that we may speak of the index of decomposition 

 of any order of multiplicity (single, double, treble, &c.) of or to 

 the modulus n. 



For single decomposition it is well known and immediately 

 obvious, that the indices to the successive moduli given by the 

 rational numbers in regular progression will be the coefficients 

 of X, x^y a,^, &c. in the continued product, 



(l-.r)-' . (l-.r2)-' . (1-*'^)"', &e. ad inf.; 

 calling these »,, n^, n^, &c., it is of course obvious, as Mr. Cayley 

 has observed, that the indices of double decomposition to the 

 same successive moduli will be the coefficients of the same argu- 

 ments X, x^, x^, &c. in the continued product 



(1-^)-"' . (l-a^2)""^ (l-a;^)-'^, &c. ad inf; 

 and by aid of this formula he has calculated (with extreme faci- 

 lity) the indices in question up to the modulus 11, and found 

 that they form the series 1, 3, 6, 14, 27, 58, 111, 223, 424, 817, 

 1527, which accordingly is the series representing the number 

 of singularities capable of existing between quadratic loci com- 

 mencing with 1 and ending with 11 variables. 



The values of n■^, n^, n^, . . ./?,,, &c. themselves are given in 

 Euler's introduction, and are respectively 



1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, &c., 

 which numbers will accordingly represent to their respectively 

 corresponding moduli the number of classes of singularity, 

 whether these classes be defined with reference to the dififerent 

 modes of distribution of the total collective singularity about 

 different points, or with reference to the degree of the lowest 

 system of minor determinants of the matrix to the determinant 

 to U + XV having one or more factors in common, which latter 

 is the mode of forming the classes adopted by me in the " Enu- 

 meration." 



Let me be permitted to express the satisfaction which I have 

 felt in finding this theory, which appeared to be doomed to hope- 

 less oblivion, thus unexpectedly, after three years of interment, 

 coming back to life, and at once filling a desired place in ana- 

 lytical researches pursued with apparently a totally different aim, 

 26 Lincoln's Inn Fields, 

 March 10, 1854. 



