506 The Rev. T. R. Robinson's Experimental Researches on the 



and as each term of (4) has a coefficient, whose law of derivation is 

 2w+1.2ra-l 



n.n + 2 ' 



we have 



y n + 2 — tin X — jr X 



n+2 f ' 



so that the successive integrals can be computed with facility. This expression, 

 however, is not so well adapted as that for a spiral, to the process oi summary 

 computation which becomes desirable when b is near r. The (n + 2a;)'* term 



n -n ^^''' ??-l.n + l ■■■?? + 2,3; -3 



whence, putting as before /> = — —, 



logQ„.,. = Q„ + log/)X.t + log(^l-^j+log(^l-^) ... .+ 



logfl %- 



Developing the logarithms, and stopping at —5, this becomes 



log. Q„.2x = log. (Q„ X />") - modulus j— --^(2^-1)+^ (4a-- - 3.r + 2) ( , 



which for a; = 10 or 20 is sufficiently rapid. 



The intermediate terms in this instance are more easily obtained by the 



method of quadratures, their sum being Qndx. This process gives 



r n 



5ia.2+a.4-.- + a.2xj = a„S^'-^+Sj, (9) 



in which 



t , i?. log p x^ losr^ p „ 



A' = x± — ^+ — jf-^±&c. 



-B'=^|±3 + 2.rlog/,! 



(7 = j!+3-2,rlogp + 10A-i, 



