1898-99.] Dr Muir on a Single Term of a Determinant. 477 
Moves (and their connection with inverted-pairs) : — 
§ 28. Convention as to mode of counting the number, p, of moves. 
§ 29. Relation between inverted-pairs and moves : fA = ^. 
§ 30. [a, still = $, when moves are made in reverse order. 
§ 31. Lowest possible number of moves necessary. 
Interchanges (and their connection with inverted-pairs) : — . 
§ 32. Convention as to mode of counting the number, v, of interchanges : 
effective and ineffective interchanges. 
§ 33. Greatest possible number of interchanges ever necessary. 
§ 34. Effect of an interchange on the number of “ inverted-pairs.” 
Circular Substitutions (and their connection with interchanges) : — 
§ 35. Mode of decomposing a substitution into circular substitutions. 
§ 36. A circular substitution from the point of view of its notation : defin- 
iteness of the number of circular substitutions for each permu- 
tation. 
§ 37. Effect of an interchange of two elements in the same circular substi- 
tution. 
§ 38. Equivalence of Cauchy’s rule to the rule of interchanges. 
§|39. Number of effective interchanges necessary. 
§ 40. Effect of an interchange of two elements in different partial circular 
substitutions. 
§ 41. Smallest number of interchanges required in any given case. 
§ 42. Reason for using effective interchanges only. 
§ 43. Retrospect. Cauchy’s investigations regarding circular substitutions 
referred to. 
§ 44. Relation of the circular substitutions of a permutation to those of the 
conjugate permutation. 
§ 45. Reason for similarity of sign of conjugate permutations. 
§ 46. Character of the circular substitutions in the case of self-conjugate 
permutations : number of the latter. 
Even Circular Substitutions : — 
§ 47. Deduction of Jenkins’ rule from Cauchy’s. 
