Macfarlane — Differentiation in the Quaternion Analysis. 205 



The next step is to investigate the differential of a product of 

 vector logarithms. On the one hand 



igA + B = eA + £ d{A + £) 



= jl+^ + ^ + (£±^\ \cl{A + £) 



= dA + dB 



+ [AdA + AdB + BdA + BdB] 



+ ^ {A'dA + AhlB + 2ABdA + 2^^f/5 + B-'dA + ^2^^} 



+ &c. 

 On the other hand 



j^ A^ AB^ B^ 



^ (3l ^ TT ^ TT "^ 3"! 



+ &c. ; 

 therefore 



d{e^eS) = dA + dB 



+ {AdA + d{AB) + BdB\ 



+ ^ [AHA + d{A^B) + f/(^^*) + BHB] 



+ &c. 



Hence, by comparison of the two results, 



d{AB) = AdB 4 BdA, 



d(A'B) = 2ABdA + A'dB, 



d{AB^) = BHA + 2ABdB. 



The factors are not differentiated in situ, as is done by Hamilton and 

 Tait ia the case of the products of ordinary vectors ; on the contrary, 

 the differential is always written at the end, for the natural order of 

 the logarithms is -4, B, dA, dB. 



