of Edinburgh, Session 1883-84. 
607 
12 + 22 32 42 + 52 
/y* /y»2 /y*3 /y»4 /y»5 
tA./ cv tA^ tAj */0 
+ ^ + &c„ 
we observe that this series is convergent for every value of less 
than unit, and even so for unit itself ; but that for any value of x 
greater than unit by however little, its terms ultimately increase so 
that their sum either is infinite, or represents an unreality. It is 
thus a moot question whether just at the limit x=\, <fiX be or be 
not infinite. 
On taking the differentials we get 
so that we shall have the required sum, if we can put this integral 
in a concrete form. 
Having constructed a logarithmic curve with AB for its base and 
AO for the length of its subtangent, take in it some point C and 
draw CE parallel to BA, then EC is the logarithm of AE. Hence 
if we place the origin of co-ordinates at 0 and denote OA by unit, 
OE by X, we have CE = - log (1 - x). 
If now we continue a straight line OC to meet AB in D, and 
draw DF parallel to AO to meet the continuation of EC, EF = AD 
becomes ■ ^ . If C be carried along the logarithmic curve, 
X 
the corresponding point F will trace the curve shown in the figure. 
When C comes to 0, the secant OC merges into the tangent, equally 
inclined to the abscissae and ordinates ; hence the distance OG must 
be equal to OA. When C passes to the other side of 0, the curve 
continues above G still nearing the interminate line AOM without 
ever reaching to it. The curve has thus two asymptotes AB and AM. 
The surface EOGF thus represents the integral 
-log(l -x) 
X 
and thus we have 
8^ and ^ 1 =AzM 1 Li£), 
y 0 X 
