584 Proceedings of the Royal Society of Edinburgh. [Sess. 
TABLE OF CONTENTS.* 
(1) Introduction . . . . . . . • • 503 
(2) The Laws, Symbols, and Parts of Multenions ..... 504 
Fundamental Laws ........ 505 
Fictits — Definition of Mnltenion ....... 505 
Fictor, Fictorplex, Fictor-product, Continent Fictorplex .... 505 
Multit, Multiplex, Order of Multiplex ...... 506 
Multiplication Combinations of Multenions ..... 506 
Conjugate, Reversate ........ 509 
Re-, Retro-, and Pro-placements ....... 509 
{3) Permissible Simplifications ....... 510 
Case of Quaternions . . . . . . . .510 
Sign of square of fictit . . . . . . . .511 
Law A : — ti 2 = f2 2 = . . . .=±1 ....... 511 
Self-Conjugate and Self-Reservate . . . . . .512 
(4) The Rigid Replacement g( )q~ 1 . Generalisation of Fictit . . . 513 
The meaning of a -1 . . . . • • . .513 
Theorems connected with An = qi n q~ 1 (w = l, 2, 3 . ... n) . . . 516 
Proof of existence of q in qp—p'q ....... 517 
Introduction of the Multilinity ....... 518 
When n is even, q~ l is finite ....... 520 
(5) Fictor Products, Complements ....... 521 
The Combinatorial Part of the fictor-product ..... 522 
Algebraic analogies in determinants and rectangular arrays . . . 523 
The Complement of a multit and of a multenion ..... 523 
Comparison with quaternions ....... 526 
>{6) A Multenion in terms of given Fictors . . . . . . 527 
Various forms of the expansion ....... 528 
Introduction of the differential operator v corresponding to a variable fictor . 531 
Symbol K replaced by vertical stroke | . . . . . . 532 
(7) Fictorlinities and Multilinities ....... 532 
Fictorlinity defined in the form 0p = 2)8SaKp = 2j8Sa|p .... 533 
Definition of Conjugate <p ' : Sp|0<r = S<r|0'p ..... 533 
The w-tic satisfied by <p . . . . . . 535 
Discussion of the roots ........ 536 
Extension of certain of the fictorlinity theorems to multilinities . . . 538 
Colinities .......... 541 
Skewlinities ......... 542 
Rotational linities ......... 544 
(8) Replacements. The Fictorlinity Replacement ..... 546 
Proplacement ; Retroplacement ; Unireplacement .... 546 
Fictorlinity Replacement is the generalisation of homogeneous strain to space 
of n dimensions ........ 550 
Ordinal Multilinities ........ 550 
The most general replacement ....... 552 
(9) Complementary Replacements, or the principle of Duality . . . 553 
Development of q in terms of products of an even number of factors . . 558 
* I have prepared this brief epitome of what seem to be the salient points, to facilitate 
reference. — C. G. Knott. 
