THE VALUES OF -^/V'V*. v»81 



_2 



V7TJ 



. 167 M 679 Q 75 Q 763 „ 97 ^ 167 



296 851 ' 94 645 ' 82 296 ^W ' 



„ 455., 31 ., 645 . 82 , 661 Q 300 31 



E^k^M, or^M = ^A, or--A= ^S=, 1(rmr or 



954 ' 65 763 ' 97 980 629 JW 65 ^W ' 



a 408, T 70 _ _ 851 . 94 . 298,-, 169 

 S = p^rM, or ttttM = ^A, or == A = kt^-E = 



577 ' 99 679 ' 75 201 239 ^ W " 



2, H3 7 1 53 _5 



M 2 355A 2 ° r 22A 2 ~ 2S 2 ~ 233E 2 ° r 22E 2 ' 



23. Besides p, other values of t corresponding to certain definite values of H may 

 occasionally be required,* and the extent of the table now given will enable us to 

 determine them with a high degree of accuracy by simple interpolation f; thus : — 



o-i, 



# = 0088 885 991 



. . log 2-948 832 9230 



. . 0-186 367 523.p 



0-2, 



0179143 455 



1-253 200 9459 



0-375 512 978./) 



03, 



0-272 462 716 



1-435 303 8936 



0-571 272 788. p 



0-4, 



0-370 807 149 



1-569 148 0986 



0-777 377 028./) 



05, 



0476 936 276 



T678 460 3565 



1-000 000 000./) 



06, 



0-595116 079 



1-744 6016843 



1-247 789503./) 



07, 



0-732 869 079 



f-865 026 3985 



1-536 618 445.p 



0-8, 



0-906193 802 



1-9572210875 



1-900 031 193.p 



0-9, 



1-163 087153 



0-065 612 2587 



2438 663635./) 



I "> co co co 



Construction of the Table. 



24. In both divisions of the general integral the factor e - ' 2 forms a multiplier. 

 Assistance in obtaining the values of this factor might have been derived from the 

 extensive tables of e~ x by Prof. F. W. Newman and Mr Glaisher,J had they been in 

 existence when the following table was begun. But the interpolation for values of 

 e~* , by means of the formula — 



«-*"=«- { 1± i + r2 ± H3 +et0 -} (40) 



is somewhat laborious, since h in this case has the form of 2xh+h 2 . As the factor in the 

 function H is the multiple —r^e' 1 , it is occasionally convenient to find its value 

 logarithmically, and also as part of the computation of the value of the function, the 

 former proving a check on the working for the latter. § In the first part of the 



* Gauss, Bestirrvm. d. Genauigkeit d. Beobacht., § 2 ; Werke, Bd. iv, S. 110. 



t Or, the difference formula (38), given above, § 20, may be used to find these values. 



t Trans. Comb. Phil. Soc, vol. xiii, (1883), pp. 145-272. 



§ If we compute in succession, as is naturally the easiest method, the terms of the expression — 



£p+-'-/+S+S-fi-{r>5+S-~> 



he sum of the 1st, 3rd, 5th, 7th, etc., terms will give the value of —6"' ! i whilst the sum of the quotients of the 2nd, 

 th, 6th, 8th, etc., terms, divided respectively by 1, 3, 5, 7, etc., will give the value of H. 

 VOL. XXXIX. PART II. (NO. 9). 2 U 



