90 G. H. KNIBBS. THE THEORY OF THE 



the final reading, see (2), being : — 



na-sec & 1 (pi+p s + ...etc) + sec/3 2 (p 2 +p± + etc), (8). 



Evaluating the probable errors of the terms after na in (8), that 



of the former term is + &Qcfi x p o >/n (9) and of the latter 



similarly + sec /3 2 p o Jn (10) ; the probable error of the sum 



of which is the square root of the sum of their squares, viz.: — 

 + V(sec 2 f$ x p o 2 n + sec 2 2 p o 2 n) = + p o */ n*/(sec~ p l +sec 2 j3 2 )... 

 (11), and this divided by n, the number of measures gives the 



probable error of the mean result : — + ^*/(sec 2 /3 x + sec 2 /3 2 )...(12), 



becoming identical with (5) when /3i and /3 2 are each zero. 



The practical significance of the result given is this : — When 

 applying an evaluation of the average probable errors of pointing 

 for some particular instrument and observer, to the discussion of 

 specific cases, the element of altitude, if considerable, should not 

 be ignored. 



Errors which may be classed as errors of pointing. 



The errors which may legitimately be included as errors of 

 pointing, in addition to that arising from visual incapacity to 

 appreciate when the cross-wires are exactly upon the image of the 

 object, (an incapacity that varies with the delicacy of those wires, 

 with the keenness of the observer's sight, with the state of the 

 atmosphere, with the quality of the definition, and, to a large 

 extent, with the magnifying power of the telescope,) are errors 

 occasioned by irregular refractions of the rays from them. These 

 operate by leading to the adoption of a false direction through 

 temporary apparent displacement, and by making it difficult to 

 decide the mean position of the (sometimes) apparently rapidly 

 vibrating object. All these errors, intrinsically identical in their 

 consequences, are in general inseparable, and can be differentiated 

 for the purpose of estimating their relative magnitudes, only by 

 discussing a very large number of measures, in which they severally 

 exist, with a similar number in which they are severally absent. 



Errors of Reading and Graduation. 



The terms in (2) assume perfect circle graduation and reading, 

 a purely hypothetical condition. Entering now upon the con- 

 sideration of errors in these elements, it may be remarked that 

 accidental errors of reading, and accidental errors of graduation 

 may indifferently be classed together as errors of reading or of 

 graduation.* Periodic errors either in the verniers or reading 



* For the purpose of this discussion. The former element probably 

 varies with each observer, more particularly when verniers are used. 

 After therefore a combined error of this character is investigated, and 

 its magnitude ascertained, it should be remembered that that magnitude 

 is true only for a particular observer and instrument. With reading 

 microscopes the graduation element is probably the greatest, next to this 



