INACCURACY OF SOME LOGARITHMIC FORMULAE. 175 



Replacing, in [17], « by 1 — ^ f c (see [14] ), and therefore (1 — <y) f — cby ^, 

 we obtain finally the required development ; viz. 



f-U = 2i-!r-c+ V^=~l{(l-9fc)... + (Lzlif)!.... I [NoteB.] [18] 

 Having advanced thus far, it will now be easy to fulfil our original intention. 



General division. 



I. To develop y in terms of a and x. 



Let « = f « [19] 



Then by [10], a' or (f ^)' = {{x{2ir + 6} 



Butby [9],2iT + ^ = f~'f^or (see [I9])f"'a 



Hence a, or (see [!])«/ = f (^f "^ «) [Note C] [20] 



Hence, expanding f (j; f~' a) by formula [7], we obtain, 



j/=l+\/ — la:f a... + ^ — i L^'J 



II. To develop x in terms of a and y. 



Solving [20], we obtain, ^ = _^^ [Note D.] [22] 



f a 



Hence, developing by formula [18], 



X = ~ ==^ J- 



2i^-c+ ^~^-f{l -afc) ... + -^{l-a{c)''...\ 



(i and c are dotted underneath, to show that when rendered determinate, their 

 individual values may difitr from those of i and c.) 



[21] and [23] may now be compared with [2], [3], and [4]. 



Remarhs on the application of the preceding theory. 



From the foregoing principles many collateral deductions may be inferred. 

 For instance, they present a solution of difliculties and illustrate peculiarities 

 appertaining to the theory of the logarithms of negative quantities. Directed 

 to geometry, they advance into an almost uninvestigated part of analysis, by 

 conducing to trace the form and evolve the properties of curves (if figures. 



[23] 



