206 Proceedings of the Royal Irish Academy. 



This principle enables lis to differentiate the reciprocal of a vector 

 logarithm. The expression B j B is equal to 1 absolutely ; by which 

 I mean, that it reduces by means of the principle that (3 / 13 = 1, not 

 by means of ^8^ = 1. As the correct place for the differential is at the 

 end 



d(B/B) = jBdB + Bd{/B) = ; 



therefore d^/B) = - /B'dB ; 



that is ^ -T, = — ^ (iiB. 



B B^ 



We may infer that generally 



d{B^)/dB - -?^^-("+l). 



Let J/B denote any quotient of vector logarithms, 



d{AIB) = jBdA - AjB-dB. 



"We observe that the solidus symbol for reciprocal is more suitable than 

 the horizontal stroke, because j B . A and AjB involve order and are 

 not equivalent to one another. The slant stroke expresses the difference, 

 whereas the horizontal stroke does not. 



Consider now the differentiation of transcendental functions of 

 vectors in which Hamilton found an obstacle. Let B denote a vector, 

 real or imaginary, which may vary both in magnitude and axis. The 

 cosh function is the more general function, that is to say, it includes 

 the cos function as a special case. Now 



^- B^ 



cosh^ = 1 + — ; + — f + etc. 

 2 ! 4! 



^(cosh^) = (^f|j+|-|+ )dB- 



therefore d (cosh B) j dB = sinh B. 



Similarly d (sinh B) / dB = cosh B. 



When B is imaginary, it may be written •v/' - 1 Bq, and 



COS Bo = 1 - ^ + 4T ~ ^*^' 



( B ^ I 



d (cos ^0) = I - -^0 + Try - etc. I dBo, 



therefore d (cos -So) / dB^ = - sin Bq. 



