1885.] A General Theory of Mathematical Form, 399 



§§ 189 — 193. Three Modes of Compounding Systems. 



These are selected as illustrative of the way in which certain forms 

 can be derived from others. 



§§ 194 — 195. A General Method of Graphically representing a Complete 



System. 



This is effected by the use of graphical units and links only. (See 

 §§ 73-83.) 



§§ 196—210. Networks. 



We here turn to the consideration of systems of pairs, and the 

 networks they compose. 



§§ 211—221. Chains. 



A simple chain is a succession of undistinguished pairs, ab, be, cd, 

 de . . . . It may be open, i.e., having terminal units, or closed. We 

 may also have compound chains containing distinguished pairs. The 

 formation of equations by considering coterminous chains is shown. 



§§ 222—239. Groups— Circuits. 



In a group each unit is unique with respect to each of the others. 

 The pairs divide up into systems of closed simple chains called circuits. 

 Groups give rise to simple associative algebras. 



§§ 240 — 252. Groups containing from One to Twelve Units. 



In these sections the graphical and tabular representations are given 

 of all groups which have less than thirteen units. 



§§ 253 — 255. Some General Forms of Groups. 



We have here merely some obvious generalisations of certain forms 

 given under the preceding head. 



§§ 256—269. A Family of Groups. 



The investigation contained in these sections was suggested by a 

 paper by the late Professor Clifford " On Grassman's Extensive 

 Algebra." 



§§ 270—274. U-adic Groups. 



These are such that the aspects of the component collections of R 

 units, when regarded as units, compose a group. (See §§ 362 — 380.) 



§§ 275—279. Substitutions. 



In considering an array of n letters admitting of certain substitu- 

 tions, we are considering a system of n units of a definite form. 



