VECTOR ANALYSIS. 79 



when N, which is a whole number, is increased indefinitely. That 

 this definition is equivalent to the preceding, will appear if the 

 expression is expanded by the binomial theorem, which is evidently 

 applicable in a case of this kind. 



These functions of 3> are homologous with <. 



172. We may define the logarithm as the function which is the 

 inverse of the exponential, so that the equations 



are equivalent, leaving it undetermined for the present whether 

 every dyadic has a logarithm, and whether a dyadic can have more 

 than one. 



173. It follows at once from the second definition of the exponential 

 function that, if < and " are homologous, 



and that, if T is a positive or negative whole number, 



{e*}T = e T* 



174. If SI and $ are homologous dyadics, and such that 



2.$=-$, 



the definitions of No. 171 give immediately 



e 8 -* = cos $+H sin <1>, 



e - E 

 whence 



e - E * = cos 3? H sin <, 



175. If $. = 



Therefore e*+* = e* + e* - 1, 



cos{<!> + } = cos $ + cos I, 



sin { $ + "^ } = sin $ + sin "^. 

 176. 



For the first member of this equation is the limit of 



|{I4-N" 1 $} N , that is, of |I4-N~ 1 $| N . 

 If we set $=*ai+fij+yk, the limit becomes that of 



(l+N- 1 a.i + N- 1 /3.,;-}-N- 1 y.&) N , or (l + N' 1 ^) 11 , 

 the limit of which is the second member of the equation to be proved. 

 177. By the definition of exponentials, the expression 



represents the limit of 



