Computation of the Functions Go(ic), Gi(^j), and ^n{^''\/ 0- ^7 



the known value of obtained in finding Jo(i>-:), gives a check on 



mistakes in calculating X. The known value of ^/^^m+i does the same 

 service in regard to the computation of or Y. 

 8. By formula (8) 



J^{x) = ft_ft + ft-ft+ 



Hence Ji{x Ji) = p^i^ - /Sgi* + - 



Pi cos |— Ps cos — + ft cos — ■ 



+ * ( ft sm - - ft sm — + ft sm — , 



+ i(ft-ft-ft + ft + ft-ftl-ft3 + ft5+ )} 



= -^{Xi + Y,i}......,say (13), 



where 



Xi - (ft)/H-l + ftw + s) - ^ {Psm + G + ftw + r)? 

 Yi = 1 (ft j/t -M + ftirt + 7) - 2 (Psm + 3 + /^S(» + 5)) 



the summatipn being in all cases from m = to the largest value of i]i 

 which gives sensible values for (3. 



The values of Xi, Yi, computed by these formulae from the known 

 values of (3, are given in Table II. 



The computations evidently admit of a check to inaccuracy of the 

 same nature as those given in the last article. 



Another form of the values of Xi and Yi is given by 



Xi = -(ft,>j + i -ftv/i+5) + S(ftm + 3-ftj« + T)) 



Yi = i (ftsj»+i - ft/rt+s) - 2 (ftw+3 - ftj/i+r)? 

 which reduces the computation to that of the two quantities 



2(ft^,i+i - ftm+5) and ^ (Psm+s - Psm+7)) 

 so that if these be denoted by Pi and Qi 



Xi = Pi + Qi, Yi = Pi-Qi. 



This form admits of somewhat ditferent checks to mistakes. The 

 values in Table II have been computed independently in the two ways, 

 so that the writer has every confidence that they may be relied on as 

 correct. The column for Yi has also been difterenced with satisfactory 

 results. 



