of using the Theodolite. 175 



in their average. The excentricity, then, has no connection 

 with the faults or advantages of this singular method ; these 

 can have reference only to erroneous graduation. 



Viewing the matter in this light, it is perfectly obvious that 

 more confidence is placed in the reading at A, than in that at 

 B in the ratio of 5 : 3, than in either of the readings at C and D 

 in the ratio of 5 : 2. The meaning of this would easily be un- 

 derstood, if it were known that the limb towards A is more 

 trustworthy than that towards B ; but then, unfortunately, this 

 best part of the limb would need the property of ubiquity, 

 since it must change place every time the telescope is directed 

 to a new signal ! 



In my report to the Society on the merits of Mr Galbraith^s 

 reflecting circle, I shewed that the correction of errors by many 

 readers is a matter of probability only ; «.and it follows that the 

 more numerous the readers are, so much is the chance of exac- 

 titude increased. Let us compute the relative inaccuracies of 

 different systems of verniers applied to a given limb. 



Let us conceive a perfect set of divisions to accompany the 

 actual ones. The distance from any perfect division to the 

 corresponding imperfect one, will be the error of the last : 

 n being the entire number of divisions, put p, q^ r, *, ^, &c. for 

 a few of the errors ; the sum of these, /? H- ^ + r -f &c., being, 

 as usual, denoted by 2/?. It would at first sight appear, that 



the Tjth part of this, that is - 2;?, is the average error to be ex- 

 pected from the graduation. This, however, is not the case; 

 for we may imagine the whole system of exact divisions shifted 

 round a little, so as to increase or to diminish all the errors 



equally. The average -2/? thus depends upon the arbitrary 

 position of the normal divisions, as well as on the actual gradua- 

 tion. Indeed, that position may be so assumed, as to render 

 this average zero. For this purpose, we have only to shift the 



normal system forward by a distance - 2 p. The different er- 

 rors would then be 



1 1 



of which the sum is evidently zero. That is, if this method of 

 averaging the error were allowable, the mean error in all in- 



