182 PROFESSOR DE MORGAN, ON THE 



It may perhaps be worthy of note that the series 



do + a x x + a % x l + 



may be considered as v e" when v = in the equation 



V€" = a u e v x « l6 u+log * + a s e v+2logx + 



I now return to the purely algebraical question. It is in our power to 

 avoid all ambiguity in results, by simply prescribing that every symbol 

 shall express not merely the length and direction of a line, but also, the 

 quantity of revolution by which a line, setting out from the unit line, is 

 supposed to attain that direction. When this is done, I shall use a double 

 sign of equality to denote it. Thus, if we denote by (a, 9) a line of a length 

 a, which has made the revolution 9, it is allowable to write 



(a, 9) = (a, 9 + 2tt) = (a, 9 + 4tt), 



but not 



(a,9) = = (a,0 + 27r) = = (a,0 + 4tt) 



As long as we neglect this additional prescription, great care will be 

 requisite to prevent our falling into error. While exponents transform 

 lengths into lengths, and directions into directions, no great caution is re- 

 quisite : but when, as we shall presently see, an exponential process causes 

 the exponent of a length to affect that of direction, or vice versa, the follow- 

 ing fallacy of a continental analyst, mentioned by Professor Peacock in his 

 Report, is frequently likely to occur. Stripped of unnecessary details, it is 

 as follows : 



or e ~ iir3n ' = 1, an absurd result. 



The answer is very simple : if no extension of explanations be contem- 

 plated, i*W^i is not necessarily = 1, since it may have an infinite number 

 of values. If the extensions be made, and if = merely denote sameness of 

 direction, the same thing is true ; for l 2 W=i or (e 2 ™^) 2 ™^" 1 has an in- 

 finite number of values, of which one only ( € <>)2™V=i is = 1 : and the same 

 fallacy might be thus propounded ; 



^/z* = + x, y/x* = — x, therefore x = — x. 



