611 
of Edinhimjh, Session 1883-84. 
Taking the fourth part of the above 
_ 1 1 1 1 
24 22 42 62 ”^82 
and subtracting, we get for the inverse squares of the old numbers 
7r2_l 1 1 1 
8 P 32 52 72 
thus showing that the term of the series 1 - 
1.2 1.-4 
1-6 
+ &c., would be exhausted by the divisors 1 
1 - 
12^2 
The successful pursuit of this line of inquiry would show that, in 
the present case, the extension of the laws of finite equations to 
equations involving interminate series is admissible ; but it would 
do so at the expense of previously discovering those coefficients 
which were thought to have been found by the summation of the 
series of inverse powers. 
The expressions for the log cosine and log sine may be developed 
directly by a process applicable to all series of the general form. 
6x=l - G^X + C.2P - C 2 X^ + &c. 
If, for shortness, we write 
A = - C2P + C3P - &c., 
we have 
, ^ A A2 A'^ A4 ^ 
-log 6x= j + -j + j + — + (fee. 
Here, from the expansions of the various powers of A, we have 
to collect the terms containing like powers of x. 
A study of the character of these expansions enables us to write 
out the details of any one term separately from the rest. Thus, if 
we wish to get the term involving we have to consider all the 
combinations giving the tenth power ; from A we have only the one 
form ; from A2, x^^ is got in as many ways as 10 is decom- 
posable into two parts, as 5 4- 5, 6 4- 4, 7 + 3, 84-2, 9 4-1; from A^ 
in as many ways as 10 may be divided into three parts, as 4-1-44-2; 
4 4-3-1-3, and so on. Hence, altogether, the number of parts of 
which the term x^^ is composed agrees with the number of ways in 
which 10 is decomposable. 
