1907-8.] Algebra after Hamilton, or Multenions. . 585 
(10) Differentiation, Integration, Jacobians . .... 559 
When p — 2ast, v = 2iDa,., chr = Sdplv.<r ...... 560 
da/dp defined as a fictor linity = S( )|v.<r. ..... 560 
Jacobians .......... 561 
Comparison of integrals over boundaries and through spaces . . . 562 
The Hessian colinity= S( )|v.v®.. . . . . . 565 
(11) Miscellaneous concluding remarks ...... 565 
SUPPLEMENT— (continuation of § 11). 
A Multenion may be regarded as a Linity ...... 569 
Simplification of proof, given q what is g -1 ? . . . . . . 569 
Generalisation of a Proof by Cayley in Tait’s Quaternions .... 569 
Expansion of /(0), any integral function of 0 . . . . . .570 
The Principal Logarithm and Principal Square-root of 0 . . . 572 
The generalised Conjugate 0' ....... 573 
When 0' = 0, roots occur in pairs of equals ...... 575 
Most general form of a multilinity <P . . . . . .575 
A multilinity may be regarded as a multenion . . . . . .576 
Conversely, a multenion belonging to an even order multiplex may be regarded as 
a multilinity in a multiplex of half its own order . . . .578 
The alternate replacement of q corresponds to the conjugate 0' 579 
The uniretroplacement of q corresponds to the ^-conjugate 0' . . . 579 
Final conclusion as to sign of the square of a fictit ..... 580 
Quaternion, Octonion, Combebiac Triquaternion, and Quadriquaternion . . 583 
(Issued separately August 28, 1908.) 
