606 
Froceeclings of the Royal Society 
inverse powers of the odd numbers, without having been conscious 
of the fatigue of the operation. Nay, even in our advanced text 
books on trigonometry, this gratuitous assumption is used to de- 
monstrate its own truth. 
Our taking for granted ” does not cease here ; we have to con- 
sider the subsequent terms of the progression. The sum of the 
products, two and two, of all the quantities 
1111 
g2 ’ 32^2 ’ 52^2 ’ J2^2 ’ 
must amount exactly to — . Now it is well known that the 
^ 24 
square of the sum of any quantities exceeds the sum of their 
squares by twice their product, taken two and two, or that 
but we have already assumed to be — , we now assume 2 . afi 
A 
to be ~ , and the obvious result of the two assumptions is that 
^ i . Thus, by this easy process, we find that 
6 
1 = J- _L J_ 1 
6 “ 1 Y ‘‘‘ 3 V 5 V 
or that 
1 1 J_ 0 
6 “96“14 ■’'3^ ■‘‘7^ 
The same line of exposition may be continued indefinitely, and it 
now becomes clear that M. Callet has not got the numerical values 
of his coefficients from the impracticable summations of the series 
(de toutes ces series) ; but that he has deduced the sums by a totally 
different and manageable process. 
The following process for computating directly the sum of the 
inverse second powers of the natural numbers may, perhaps, have 
some points of novelty and interest. 
Denoting by the symbol (f)X, a function whose development is 
