616 
Proceedings of the Royal Society 
TT ^-u • 22 368 256 i. i . • 
Hence the term is— — — — — - — — — ^a}^ which, on being sini- 
211 . 3^ . 52 . 72 . 11 . 13 » » 
plified, becomes 
2.43.127 10922 
35.52.72.11.13“ ~ 42667525“ ' 
In this way we get the following results : — 
^lep. log sec a = 
1 9 1 4 1 
¥“ +¥73“ +¥75 
a^ + 
17 
23 . 32 . 5 . 7 
-a° + 
31 
34 . 52 . 7 
no 
691 ,, 2.43.127 
■*■2.3^52.7.11^ ■*■35.52.72.11.13^ 
257.3617 73.43867 
^2*. 36. 53. 72. 11 . 13 ^33. 53. 72. 11 . 13. 17 
31.41.283.617 ^ 
■*■2.33.54.72.11.13.17.19 
Nep. log -A- = 
sin a 
^ a2 + ^. L -a4+ v,7 -- l— + 
1 
2.3 ‘ 22 . 32 . 5 34 . 5 . 7 23 . 33 . 52 . 7 
1 691 
-^10-1 
35. 52. 7. 11 ^2. 37. 53. 72. 11 . 13^ 
2 ... 3617 
/I2 
ai4 + 
,16 
36. 52. 72. 11 . 13 24. 37. 54. 72. 11. 13. 17 
43867 283.617 
ai3 + 
341.53.73.11.13.17.19 2.39.56.72.112.13.1.7.19 
a20 + &c. 
The coefficients in the first series are multiples of the corre- 
sponding coefficients in the second series, the multiplier for the terms 
a2” being a2« _ 1 or (a” - l)(a”+ 1). Thus for Cfc2 the multiplier is 
1.3; for «4 it is 3.5; for it is 7 . 9, and so on ; which is in 
accordance with the obvious law, that the sum of the nth. inverse 
powers of the natural numbers, is 2n times that of the series of 
even numbers. 
