370 macfarlane: algebra of physics 



The new more nearly resembled ordinary differentiation, gave 

 a differential coefficient, and allowed Taylor's Theorem to be 

 generalised without difficulty. 



Brief of twelve lectures on space-analysis. University of Penn- 

 sylvania. University Bulletin, April, 1900. The imaginary expres- 

 sion ia is merely a convenient way of writing a v/2 and in reality 

 means the same thing. Using this notation, the fundamental 

 principle for quadrants is 



(ia) (t/3) = — cos aj3 — sin afi . i [a/3]. 



By dividing out ii and the equivalent — we derive 



a/3 = COS a/3 + sin a/3 . i [a/3] 



which is the fundamental principle for vectors. This derivation 

 explains the necessary presence of i in the second partial product; 

 a point which is ignored by vector-analysts. 



Vector differentiation. Read before the Philosophical Society 

 of Washington. March 31, 1900. Bull. Philos. Soc. Wash. 14: 

 73-92. The paper begins by referring to the two kinds of differ- 

 entiation, depending on the two forms of multiplication. For 

 the direct square 



d(A 2 ) = 2AdA 



» 

 and for the direct product 



d(AB) = AdB+BdA. 



But for the conjugate square 



d(AA) = dA . A + AdA, 

 and for the conjugate product 



d(AB) = dA . B + AdB. 



The latter form is the, only kind considered by quaternionists 

 and vector-analysts, and is called differentiation in situ. 



The application to the modulus and the unit was pointed out. 

 For example, if R = rp, the direct square is r 2 p 2 ; and d(r 2 ) = 2rdr, 

 d(p 2 ) = 2pdp. But for the conjugate square RR = r 2 pp; and 

 d(r 2 ) = 2rdr as before, but d (pp) = dp . p + pdp = o. Again 

 d(p z ) is simply 3p 2 dp; but d(ppp) = dp . p 2 + pdpp + p 2 dp. 



