298 Sm William Rowan Hamilton on Fluctuating Functions. 

 so that 



«2 = 1> <«4 = i. ^e = !%-• (kO 



Again, by making j^ zz a*, but supposing i = an uneven number 2k, we get the 

 following additional term in the second member of the equation (f ^)} 



(-i)*[2;cr«,.^„ (F) 



in which 



thus 



1 = w, = 2a.2 - 2. 1 tt.3 =4m2 — 4. 3. 2«., + 4. 3. 2. 1 m^, (n'') 



so that 



Wj = 1, W3 = ^, Wj = ^. (o'') 



Accordingly, if we multiply the values (k '') by — , --, t— -, we get the known 



values for the sums of the reciprocals of the squares, fourth powers, and sixth 



It 1^ 

 powers of the odd whole numbers ; and if we multiply the values (0'') by -, -t^j 



^, we get the known values for the sums of the reciprocals of the first, third, and 



fifth powers of the same odd numbers, taken however with alternately positive and 

 negative signs. Again, if we make^^ = sin a, in (e''), and divide both members 

 of the resulting equation by cos I, we get this known expression for a tangent, 



which shows that, with the notation (h''), 



tan^ = «»j^' + w4P+We^* + ...; W) 



so that the coefficients of the ascending powers of the arc in the development of 

 its tangent are connected with each other by the relations (f^), which may be 

 briefly represented thus : 



V^^\ = (14- V"^ D„)^*- tan ; (r ^ 



the second member of this symbolic equation being supposed to be developed, and 



