Theory of Indices. 23 



formula (c) could be used when n<,r 2i^ well as when «>r, and, 

 if we like, we might retain formula (c) and entirely dispense with 

 formula (d). 



Again, it may be seen, in a similar manner, that if <2~''=— form- 



ula (d) could be used when ^z>r as well as when «<r, so that 

 we might, if we like, retain formula (d) and entirely dispense 

 with formula (c). 



If we find that we may use negative exponents upon the above 

 interpretation, then we will for the most part dispense with form- 

 ula (d), using it only now and then, if at all, when it comes a 

 little handier than formula (c). 



18. Again, by the above interpretation formula (c) can be used 

 when one or both of the exponents are negative. 

 First, suppose r negative and equal to —q, then 



But substituting in (c), 

 the same result as before, so that formula (c) may be used when 



a^ * a^ a'' a' a'' «'+'"' 



Second, suppose 71 negative and equal to —s, then 



a' 

 But by substituting in (c), 



the same result as before if our interpretation of negative ex- 

 ponents be correct, so that fomiula (c) may be used when 7t is 

 negative. 



Third, suppose both 71 and r negative and let /z=— .?and r=—q^ 

 then 



a a'' a' 

 But by substituting in the formula, 



the same result as before, hence formula (c) may be used when 

 both 71 and r are negative. 



