6o 4 SCIENCE PROGRESS 



(i n ). The general problem receives extension and elucidation 

 beyond the discussion already given. The composition of 

 bipartite numbers is dealt with by a graphical representa- 

 tion. Two chapters are devoted to the following problem, 

 which was suggested by the American astronomer Newcomb 

 and which arose in connection with a game of " patience " : a 

 pack of cards containing p cards marked i, q cards marked 2, 

 r marked 3, ... is dealt after shuffling, and so long as the cards 

 dealt have an ascending order or are of equal value they are 

 dealt on one heap, but when this sequence is broken, a new 

 heap is commenced, and the process continued until all the 

 pack is dealt ; the question is, to find the probability that there 

 will be exactly m heaps. The solution leads to new analysis 

 and to a remarkable chain of theorems for which the curious 

 reader must consult the book. 



Problems which are connected with distributions upon a 

 chessboard occupy Section V, a chapter on perfect partitions 

 being prefixed. In the application of the differential calculus 

 to the enumeration of the number of arrangements ofni's upon 

 a chessboard of w 8 squares, in such wise that no row or column 

 contains more than one symbol, we have an example of the 

 method in which in this book a complex process is illustrated 

 by a simple instance, and also an illustration of the employ- 

 ment of the differential calculus in problems which deal with 

 magnitudes so distinct from its proper subject-matter. The 

 author writes the operation Dx 11 in the form 



IXXX . . . x + X1XX . . . X + . . . + XXX . ... XI, 



Thus the various terms can be represented upon n lattices of 

 n squares arranged in line, the unity occupying its proper place 

 in the series and the squares corresponding to the x's being 

 left blank. Each term is next taken and differentiated thus : 



Dixx . . . x = 1 ixx . . .x + ixix .x + ... ixx . . . XI 



Each term here can be represented on a lattice or rectangular 

 chessboard having two rows of n squares, thus one unity is 

 placed in the position it occupies in the one-row lattice and 

 the second is placed in its proper position in the second row. 

 Treating each term of Dx n in this way we require n(n — 1) 

 lattices of two rows to give the various arrangements of two 

 unities one in each row, and in no case being on the same 



