292 Major MacMahon. Memoir on the [Nov. 24, 



Section 3 is taken up with the graphical representation of bipartite 

 numbers. A reticulation is formed which consists of a series of 

 points through each of which straight lines pass in two definite 

 directions, the boundary of the whole being a parallelogram. 



The figure AB is the graph of the number 54. A composition of 

 this number is defined by fixing nodes at certain points which possess 

 the property that no point is at once above and to the left of any 

 other point; the parallelogram between adjacent nodes is the graph 

 of a certain number, and in passing through the nodes in succession 

 from A to B an ordered assemblage of numbers is found which con- 

 stitutes a composition of the number^ which is represented by the 

 whole graph. 



This conception leads to theorems of a new kind which are general- 

 ised in Section 4 to include tripartite and multipartite numbers. 

 This section is the most important part of the investigation. It is 

 established that 



+a 



+2oJ} 



is also a generating functipn which enumerates the compositions ; the 

 coefficient of 



being the number of compositions possessed by the multipartite 



The generating function of the previous section 2 may, by the 

 addition of the fraction -| and the substitution of s^i, s z x^ &c., for 

 i, 2> &c., be thrown into the form 



1 2 



