A NEW FORM OF TELEMETER. 619* 



will be to hand in the form hca/lm-, h being the measure on the 

 vertical staff and I the sloping distance. If each of these quan- 

 tities be multiplied by the sine of its corresponding circle reading 

 ^m, and similarly by the cosine, then the sum of the former (the 

 sine products) divided by that of the latter (the cosine products) 

 with its sign changed will be — 



2 V- sin. ej ^ nk sin. a 



— TTT \ ^ = tan. o (8) 



— ^0(7 ^^^- ^ ) ^n k COS. a 



in which wis 360°/^; when alms been found, Z; may be derived from 

 either the numerator or denominator of the middle member in (8), 

 these being the exact (unrediiced) equivalents of the terms in the 

 left hand member of the equation : k -\- c is \jn of the sum of hjl. 

 The adjustment is made by setting the pointer at the reading — a, 

 and, while keej^ing the axis in the same relative position, adjusting 

 the pointer to zero. 



In discussing the telemetric possibilities of the instrument there 

 are two sources of error requiring attention, viz., inaccuracy in 

 determining the constant k, or the angle i on which it depends, 

 and error of graduation. If k be carefully determined by read- 

 ings at every 45° the error in the angle ( will probably not be 

 greater than 10", and if the graduated circle be 5in. or Gin. in 

 diameter, and be read by the verniers, 30" may be accepted as the 

 probable limit of reading and graduation error. By differentiation 

 we have from (2), (3) and (4) for a small error in the assumed 

 value of the inclination angle the resulting error of grade, viz. — 



±1 = - d,2 cot. 2 . (1 -^1)....(9) 



from which we observe that, with d i constant, the ratio dgjg is 

 sensibly constant for any given instrument, and that in different 

 instruments it is greatest for that with the least axial inclination. 



For an instrument capable of measuring a grade of 1 in 2^, or 

 21° 48', the value of the ratio is — 



+ p) 



—00002424 



9' 



Since, as we have noticed, this ratio is sensibly constant, the 

 factor gg' l{g -^ g) or wi, in ( 1 ) must be multiplied by 1 ± dglg to give 

 the true distance h, so that the effect of a 10" error in the assumed, 

 instrument will not be more than -00024, or say 1 in 4000. It is 

 obvious that — 



-^=-i^ ....(10) 

 9 ' 



is sufficiently exact for comparisons of this character. 



It is a question, however, whether after wear the axis will not 

 fit loosely, so that a considerable uncertainty in the angle of incli- 



