68 



Major P. A. MacMahon 



[Feb. 14, 



stitution can take place in seven diflferent ways, it is seen that we 

 can regard the j)rocess of differentiating twice as composed of 8 x 

 7 = 56 minor processes, each of which can be diagrammatically 

 represented by two units, one in each of the first two rows of the 

 chess-board, in positions corresponding to the substitutions of unity 

 for X that have been carried out. Proceeding in this manner in 

 regular order up to the eighth differentiation, we find that the whole 

 process of differentiating x^ eight times in succession can be de- 

 composed into 8x7x6x5x4x3x2x1 = 40,320 minor 

 processes, each of which is denoted by a diagram which slight re- 

 flection shows is a solution of the castle problem (Fig. 11). There 



cA/ *X< tAi 



*K/ tX/ «Il/ 



•A^ «^ «.|l/ 



aA/ c^ «^ 



tkj A- «L« 



r-ln i-w» /J^ 



*Kl *\i tX/ 



r^ /V* r^ 



•^ «A> vK/ 



X X 





Fig. 11. 



are, in fact, no more solutions, and the whole series of 40,320 dia- 

 grams constitutes a j)icture in detail of the differentiations. Simple 

 differentiations of integral powers thus yield the enumerative solu- 

 tions of the castle problem on chess-boards of any size. 



We have here a clue to a method for the investigation of these 

 chess-board problems ; it is the grain of mustard seed which has 

 grown up into a tree of vigorous growth, throwing out branches and 

 roots in all sorts of unexpected directions. The above illustrations 

 of differentiation gave birth to the idea that it might be possible to 

 design pairs of mathematical processes and functions which would 

 yield the solution of chess-board problems of a more difficult cha- 

 racter. Two plans of operation present themselves. In the first 

 place we may take up a particular question, the Latin square for 

 instance, and attempt to design, on the one hand, a process, and, on 

 the other hand, a function the combination of which will lead to the 

 series of diagrams. In the second place, we may have no particular 

 problem in view, but simply start by designing a process and a 

 function, and examine the properties of the series of diagrams to 

 which the combination leads. The first of these plans is the more 

 difficult, but was actually accomplished in the case of the Latin square 

 and some other questions ; but the second plan, which is the proper 

 method of investigation, met with great success, and the Latin square 



