60 BULLETIN OF THE 



1 1 



but rt = 8 sin~' J-~7^ -•- 4 sin ~* " >"^~ from equation (h), which, 



when substituted in this last equation, gives 



( 2 / 1 \ 2.4/ 1 \ 2 2 4 6/ 1 \ ' ) 



"=^n^+3(ro)+3T5(ro) +3:57,(10) +*4 



( 2/2\ 2 4/2\^24 6/2\3 ) 



+ -^ni+^(Too)+^5(ioo) +3:5.7(100) +^n 



= 2.4 X + . 56 t/= 2.4(0:4-^2/), 



If we now examine the terms, we see that the coefficient of the 



n*'^ terra is deduced from that of the (/i-l)"" term by multiplying 



2/1 — 2 1 /1\"* 



it by -= 1 — ' ; therefore if we subtract the f ,- ) 



2n~l 2n-l \2n-l/ 



part of itself from each term and remove the result one figure to 



the right, we obtain the next term of the first series, &c. 



For the second series we start our computation with the same 

 number and multiply each succeeding term in the first series by 

 the consecutive powers of (j^), and lastly multiply the result by 

 /^ and add it to the first series. 



The computation in this manner gives every terra positive, and 

 by computing the result to thirty places of decimals and taking 

 every time the nearest unit in the last place, it gives the thirtieth 

 figure accurate. 



