278 REPORTS ON THE STATE OF SCIENCE, ETC. 



i.e. the divergent terms of (6) are represented by the product of epa and the nth term ; 

 92 is given by the series 



e-" 1 + 



hV 



4 "60" 



144 "^ 336/ 



A- Kb + - - — 4- — _ 1!- 4- _^'_ \ + 1 

 v*\ 5 360 252 128 3456 7 " " J 



where 60 = —0-33333 33333 33 



6j= +0-02962 96296 3 



62= +0-00282 18695 



6g= —0-00188 1246 



6^= -0-00071 196 



65= +0-00067 84 



^6= +0-00047 3 



6,= —0-00059 



6g= -0-0006 



In this case, an improvement may be effected in the approximation to the value 

 of the function M ( — ^ . ^ . x), when x is comparatively small. Even with the short 

 table of b„ given above, when x=10-5, twelve places of decimals can be added to the 

 result where the terms of the asymptotic series beyond the n'^ are neglected. With 

 more extended tables of 6„, the approximation has been carried to twenty places : 

 when X has the small value 3-5, to ten places. When a- is a negative integer, 

 M (a . Y • ^) reduces to a polynomial which is easily evaluated. 



