96 G. H. KNIBBS. THE THEORY OF THE 



+ 0*26" and this error may be employed to proceed to a higher 

 approximation of the above, q and g o , in the following way* : — 

 + 0*26" x 1*483 = +0*386" the "mean error" or " error of 'mean 

 square" of result, the multiplier being that number which expresses 

 the value of the " mean error" in terms of the " probable error." 

 Denoting this corrective quantity by the letter e, to the sum of the 

 squares of the residuals (2v 2 , and 2u 2 ) the quantity ne 2 and ne' 2 

 should be added, and formula (22) used in both instances for dis- 

 covering the probable errors, which are, seeing 19 x(0*386) 2 = 

 2-83, and 10 x (0-386) 2 = 1-49, -675 .v/{(1666*51 + 2*83)-f- 19} = 

 + 6-33" and -675 v{(8*50 + 149) 4- 10 j = +0-675". Introducing 

 these new values in the above equations, 39*69" and 43*56" become 

 respectively 40*07" and 45*56", and resolving with these, q- is 0*61" 

 q is 0-78", g 2 19*73" and g o 4*44".f 



By (24) p o = q (25); so that in the 



V(sec 2 $ x + sec 2 /3 2 ) 

 present example, where j3 x and /3 2 are each zero, it is -^, and this 



numerically determined from the value of q last found, is 

 + 0*55". The former value gives + 0*46". The evaluation of 

 such errors, by a discussion of so few a number of measures as 

 nineteen, can of course be expected to lead to results by no means 

 of a highly approximate character ; yet a number of examinations 

 of the performances of the same instrument, of a varied as well 

 as of a similar character to the preceding, sufficiently confirmed 

 the accuracy of the values discovered by it. J 



System of deducing most probable result. 



The liability of a result given, as in I., by dividing the (n+l)th 

 reading by n to the prejudicial effect of any abnormity in the two 

 reading errors, on which it depends, has already been adverted to. 

 In the method shewn in II., each quantity is affected by twenty 



# The ' mean errors ' of residuals will be the ' mean errors ' of the mean 

 results employed (or of 82° 59' 52*7 // in the example) and therefore if they 

 be denoted by e v and e u the probable errors of a single measure and of 

 a single ten-measure mean are _+ *675 A /'2'" 2 +ne v 2 and + *675 v*^ 2 + n '*u 2 



n n' 



e-y and e u being equal in this particular instance. e u = 1-483 r u if r u be 

 the probable error of a residual, or ru . 



•675 



f The preceding note applies here also. The whole must be viewed 

 simply as an illustration of method, and as hereinafter the same principle 

 will be followed, the same qualification should be understood. 



X The values when obtained will not be general, but true only for a 

 particular observer, as the pointing error, and the reading element of 

 the reading and graduation error will vary with the observer, and with 

 the observer's physical condition. This is conspicuously the case with 

 unpractised observers. ' 



