608 
Proceedings of the Boyal Society 
and the entire surface AOGF B is the value of the series 
p + 4 + p + &c., as measured in squares of the unit OA . 
If, in the expression 
S.</>*= -log(l -*)•“ 
we substitute 1 - z for a? , we have 
8 , -x) = - log X . ^ , 
whence 
8[cf)X-\-cf>{\ -x'^ = - log (1 - cr) . 8 log a? - log . 8 log (1 - a;) ; 
which, on be integrated, becomes 
(f>x + (f>{l - ir) = C - log a? . log (1 - a?) . 
Having made Oe equal to AE or 1 - Xj draw ecf parallel to 
EOF j then is the surface eOGf the representative of </>(!- ir) , and 
EOGF + eOG/+EC.ec = C. 
Again, if through K , the middle of OA , we draw KHL parallel to 
AB , the equation takes the form 
2.KOGL + KN2 = C, 
so that the nnmerical value of the constant of integration is easily 
computed; it is represented by the algebraic symbol 
C = (log 2)2 + 2 1 + &c. I . 
Each term of this progression is less than half of the preceding 
term, so that the convergence is not slow. When the computation 
is carried to a considerable number of places, the convergence is so 
nearly by halving, that the sum of the terms beyond the last com- 
puted one may be held as nearly a repetition thereof. 
The following is the couiputation to ten places : — 
