of Lines and Surfaces of the Second Order. 333 



total number of decreasing* series of positive integers, com- 

 mencing with a given number (r), subject to the condition that 

 the second diflPerences shall be all positive ; which (he adds) call- 

 ing the successive second differences 8, 6', B", &c., is tantamount 

 to finding the number of ways that the equation ?•=8-^28'^-38" 

 -f- &c. admits of being solved by positive integers, which is 

 obviously the same as the number of modes in which r admits 

 of being decomposed into positive integer parts. Thus the idea 

 of partition, which arises naturally in the first part of the pro- 

 cess (that, namely, of the decomposition of the collective inclu- 

 sive-singularity in every possible way into modes of distributive 

 inclusive-singularity), reappears quite unexpectedly (it may almost 

 be said miraculously), and as the result of an analytical trans- 

 formation in the second part of the same. 



It should be observed that the case of complete coincidence 

 between U and V, which, supposing them to be functions of (n) 

 variables, corresponds to the supposition of the same factor oc- 

 curring respectively n times, (n — 1) times, {n—2) times, &c. 

 2 times and 1 time in the complete determinant, the 1st minor 

 system, the 2nd minor system, &c., the (« — 2)th minor system 

 and the (w — l)th minor system respectively, is here taken as the 

 highest case of singularity; this and the case of non-singularity, 

 which also adds a unit to the index of singularity, properly so 

 called, will together make a difference of 2 units in the numbers 

 given by me in the paper referred to, which numbers will accord- 

 ingly be 3, 6, 14, &c., in lieu of 1, 2, 12 f, &c. We are now 

 enabled to give the following simple statement of the law for 

 determining the total number of singularities which can exist 

 between two quadratic forms of n variables (or if we like so to 

 say, more generally between two linear substitution-systems of 

 the nth order, viz. the number of the singularities (including abso- 

 lute unrelatedness and entire coincidence within the purview of 

 the term) is the index of double decomposition into parts of the 

 number n. To raise up in the mind a clear conception of the 

 idea of double decomposition, we may proceed as follows : — 

 First. Suppose a state of things in which a body is supposed to 

 be determined completely, provided that the number of mole- 

 cules which it contains, and the different number of atoms in 

 each molecule are given, the index of simple decom])osition, i. e. 

 of ordinary partitionmcnt of the number of «, will be the number 

 of different bodies which arc capable of being formed out of n 

 atoms. Now imagine that, for the complete determination of a 



* Such a series must, from its very nature, be always ileerciising or 

 increasing in the same direction. 



t These numbers refer to quadratic homogeneous functions, containing 

 respectively 2, .'i, 4, &c. variables. For the case of functions containing 

 but one variable there is no distinction between coincidence and unrelated- 

 ness, and the number of modes of relation is a single unit. 



