100 



G. H. KNIBBS. THE THEORY OF THE 



opposite the line representing the relative value of g to p, by the 

 absolute value of p divided by 100. 



III. 



Table of values \ 



V(2 



Q.2 



+■ 2np 2 ' 



) Formula 



(16). 





p=100 



Number of Repetitions, n 



g 



2 



3 



4 



5 6 



7 



8 



9 



10 



15 



20 



25 



030 



30 



027 



10C 



014 



200 



173 



125 



100 



085 075 



067 



061 : 057 ' 053 \ 041 



035 



250 



203 



143 



113 



095 | 082 



074 



C67 ■ 061 ! 057 



043 



036 



032 



028 



015 



300 



235 



163 



125 



106 1 091 



081 



073 067 '• 062 



046 



038 



033 



029 



015 1 



350 



267 



184 



143 



115 1 101 



089 1080 073 i 067 



049 



040 



035 



031 



015 ! 



400 



300 



205 



158 



130 ! Ill 



097 j 087 ; 079 J 072 052 



042 



036 



032 



015 



450 



334 



227 



174 



142 1 121 



105 , 094 ! 085 ! 078 | 056 



045 



038 



034 



016 



500 



367 



249 



190 



155 | 131 



114 



102 | 092 ■ 084 060 



047 



040 



035 



016 



550 



402 



272 



207 



168 1 142 



123 



109 1 098 1 090 



063 



050 



042 



037 



016 



600 



436 



294 t 224 



181 153 



131 116 1 105 096 



067 



053 



044 



038 



016 



650 



470 



317 i 240 



194 164 



142 : 125 | 112 102 



071 



056 



046 



040 



017 



700 



505 



340 ■ 257 i 208 ; 175 



151 j 133 | 119 i 109 



075 



059 



049 



042 



017 



750 



540 



363 274 1 221 1 186 



161 142 1 127 i 115 



080 



062 



051 



044 



018 



800 



574 



386 j 292 ! 235 ! 197 



170 



150 134 | 122 084 



065 



053 



046 



018 



850 



609 



409 | 309 i 248 



208 



180 



159 ! 142 ! 128 ' 088 



068 



055 



048 



019 



900 



644 



432 I 326 ! 262 



220 



190 167 1 149 i 135 1 093 



071 



058 



050 



019 



950 



679 



455 ! 343 



276 



231 



199 ! 175 ! 157 1 142 097 



074 



060 



052 



020 



1000 



714 



478 ; 361 



290 



243 



209 - 184 1 164 ; 148 ! 101 



077 



063 



054 



020 



1500 



1065 



712 ! 535 



429 



358 



308 1 270 240 1 217 : 147 



111 



089 



075 



025 



2000 



1 1418 



946 l 711 569 475 407 ' 357 ' 318 ' 286 1 192 145 



117 



098 



032 



Limit of advantage of the reduction system of II. 



The repeated unequal inclusion of some of the errors of point- 

 ing, in order to utilize each reading in finding the best value for 

 the angle measured, cannot be deemed theoretically satisfactory ; 

 and the extent to which this is advantageous is consequently a 

 proper subject of inquiry. 



Putting 2m— 1, as in the Rule, for the number of observations, 



it may be shewn that the general expression for (28) is q v ™~ + '- 



...(29), the quantity under the radical sign being the sum of the 

 squares of the series of numbers 1 to m to 1) and as this is to be 

 divided by m 2 , the probable error of the p terms in (26) will be 



found on reduction to be + q *J ^ m8 (30). That of the same 



terms in the sum of the repeats, as shewn in I., is^? o y[2(2m - 1)}, 

 and p o v2 being equal to q, this term becomes when divided by 



2m- 1 the number of measures, q Vkz—i (31)- (30) and (31) 



are an equality when m = l; but when m is more than 1, (30) is 

 the greater. There may therefore be a limit to the advantage of 

 employing the rule exhibited in II., viz., that determined by the 

 balancing of the increase of probable error through the excess of 

 (30) over (31) with the decrease of the same arising from the 

 inclusion of all the readings in the mean result. Such a limit, if 

 it exists, is clearly dependent upon the relative magnitudes of g 

 and p. The necessary criterion is thus established : — 



