INACCURACY OF SOME LOGARITHMIC FORMULAS. 
175 
Replacing 1 , in [17], » by 1 — 6 f c (see [14] ), and therefore (1 — u) f — cby 6, 
we obtain finally the required development ; viz. 
f -1 0 = 2in-c + —OFc)... + ^ } [ NoteB -l [ 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 a = U 
Then by [10], a or (f 6f = f {x (2 i* + 6} 
But by [9], 2 in + 6 = f 1 F6 or (see [19] ) f ~ X a 
Hence a\ or (see [1]) y-F{xF 1 a) [Note C.] 
Hence, expanding f (x fi " 1 a) by formula [7] , we obtain, 
y = i + v' 
1 xf 1 a . . 
+ 
1 xF 1 a) 
1 . 2 . 
[19] 
[ 20 ] 
[ 21 ] 
II. To develop x in terms of a and y. 
f _1 
Solving [20], we obtain, x = __ [Note D.] 
f a 
Hence, developing by formula [18], 
2 jar — c+ - yFc ) ... + (1 -yFc) n ...^ 
2 in — c + V — 1-^(1 — aFc) ... + -^-(1— afc)”...^> 
(i and c are dotted underneath, to show that when rendered determinate, their 
individual values may differ from those of i and c.) 
[21] and [23] may now be compared with [2], [3], and [4]. 
Remarks on the application of the preceding theory. 
From the foregoing principles many collateral deductions may be inferred. 
For instance, they present a solution of difficulties 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, 
[ 22 ] 
[23] 
