and the general Theory of Compound Partition. 375 



ThuSj by way of veiy simple illustratiorij suppose it required 

 to find in how many ways the number m can be made up of yu. 

 elements, limited to consist of the numbers 1, 2, 3. My method 

 gives me at once the following solution. Call v the number 

 required. Then m must be not less than fx, and not greater 

 than 3/x., or there will be no solutions. For all values of m be- 

 tween [jb and 3yL6j both inclusive, 



for all values of m between 2/u, and 3/i, still both inclusive, 



It will be observed that when 7n = 2/j,, the two formulae give the 

 same value, so that either may be employed. Again, suppose 

 we wish to express the number of modes of composition of in 

 with the four elements 1, 2, 3, 4, the number of parts being fi, 



— must be not less than 1 nor greater than 4, or there will be 



no solutions possible. 



For all values of m from /j, to 2j«, inclusive, 



p, p' being the prime cube roots of unity. 

 For all values of m from 2//, to 3/x inclusive^ 



_ (m-/^-f3)^ _ (»i- 2/^-3)^ 73 

 "" 12 4 "^36 



Finally, for all values of m from S/j, to 4/j, inclusive, 



At the joining points (so to say) between the successive cases, 

 viz. where m — 2fj. or m = 3/A, the contiguous formulae give like 

 results whichever of them is applied, so that the discontinuity 

 in the form of the solution resembles that arising from the 

 juxtaposition of different curves*. This discontinuity (in itself 



* The connexion between the contiguous formula is ahvays closer than 

 what is symboHzed by the j)hrase used above. Tlie curves must be re- 

 garded asnot merely jjlaced end to end, but to lie, as it were, knit or 

 spliced together through a certain finite portion of the extent of each of 

 them. Thus the first and second formulae in the text coincide in value, 

 not merely for m + 2/x, but also for »i=2^— 1 and m=2/i— 2; and the 

 second and tiiird formulic coincide, not merely for m='Aix, but also for 



