of Edinhurgh, Session 1883 - 84 . 617 
The law of formation of the coefficients in these series may he 
shown in the following manner. If the polynome 
<px = 1 1 - - &c. 
be the continued product of factors 1 -r'a?, l-r"x, l- r"'x, &c., 
and if we multiply each term by its exponent with the sign changed 
so as to get 
c-^x - 202^:2 + ^c^x^ - ic^x^ + &c. 
and divide this by the original polynome, developing the quotient 
in an interminate series, 
d-^x + d^x^^ + d^x^ + d^x^ + &c. 
the coefficient d-^ is the sum of the /, r", r'", d^ is the sum of 
their squares, df = 2 ?’^, and so on ; wherefore 
^d-^x + ^d^x"^ + ^d^x^ + ^d^x^ + &c. 
is the logarithm of their continued product, that is the logarithm 
of ^x. Now, the coefficients , d2 , d^ are formed from the suc- 
cessive equations 
0 
0 
0 
0 
and so on. 
The application of this general method to our present series gives 
the coefficients with much less labour than the process previously 
detailed, because each result is useful in the subsequent steps ; but 
the former has the advantage of independent computation. The 
one operation serves as a needful check upon the other. 
Converting these coefficients into decimals, we have 
Nep. log sec a = 
•50000 00000 00000 00000 X 
+ 8333 33333 33333 33333 x 
= c-^-d-^ 
= - 2(^2 - r ?2 
= + 3cg - cZjC2 + d^c-^ - d^ 
4^4 ^1^3 ^2^2 *b ^3^1 d^ 
