and the general Theory of Compound Partition. 373 



going into any details Thus, then, may the theorem be stated 

 n general terms : — 



An7j given system of simultaneous simple equations to be solved 

 in positive integers being proposed, the determination of the num- 

 hei' of solutions of which they admit may in all cases be made to 

 depend upon the like determination for one or more systems of equa- 

 tions of a certain fixed standard form. When a system of r equa- 

 lions betiveen n variables of the aforesaid standard form is given, 

 the determination of the number of solutions in positive integers of 

 which it admits may be made to depend on the like determination 



for — = — Y\ single independent equations derived 



from those of the given system by the ordinary process of elimina- 

 tion, ivith a slight modification ; the final result being obtained by 

 taking the sum of certain numerical multiples [some positive, others 

 negative) of the numbers corresponding to those independent de- 

 terminations. This process admits of being applied in a variety of 

 modes, the resulting sum of course remaining unaltered in value 

 whichever mode is employed, only appearing for each such mode 

 made up of a different set of coijijjonent parts^. 



In the Problem of the Virgins, where but two equations are 

 concerned, the equations are reduced to the standard form when 

 the two coefficients of every the same variable in the two equa- 

 tions are prime to one another, and when no two pairs of coeffi- 



* Since the above was in print, I have discovered a much more specific 

 theorem, which, indeed, is to he regarded as the fundamental theorem in 

 the doctrine of compound partition, and the basis of that given in the text. 

 It is as follows : — If there be r simultaneous simple equations betiveen n 

 variables {in which the coefficients are all positive or negative integers) 

 forming a definite system (i. e. one in which no variable can become indefi- 

 nitely great in the positive direction without one or more of the others be- 

 coming negative), and if the r coefficients belonging to each of the same 

 variable are exempt from a factor common to them all, and if not more than 

 r — 1 of the variables can be eliminated simultaneously between the r equa- 

 tions, then the determination of the number of positive integer solutions of 

 the given system may be made to depend on like determinations for each of n 

 derived independent systems, in each of which the number of variables and 

 equations is one less than in the original system. 



This reduction in general can be effected in a great but limited variety 

 of modes. When only two equations, however, are concerned, the number 

 of modes is always two, neither more nor less. So that in fact we are still 

 navigating in tlie narrows, and have not fairly entered upon the wide ocean 

 of the theory of comi)ound j)artitions until we have passed the case of 

 double partition. When the given system supposed deiinite is one of three 

 equations between four variables, the number of modes of reduction is 

 twelve or sixteen, according to that type out of two (to one or the other 

 of which it must of necessity belong) under whidi the system falls. The 

 theory of ty])cs a])plicable to any system of simultaneous simjile equations 

 with rational coefficients, here faintly shadowed fortii, constitutes, I appre- 

 hend, a new and important branch in the theory of inequalities. 



