620 PROCEEDINGS OF SECTION H. 



nation may be expected. Probably 1 in 2000 -would be as much, 

 as one ought to expect after wear in the supposed instiument. 



With regard to error of graduation or of reading, we find the 

 ratio of the grade error to the grade to be 



— ^ = — rf^cot. ^^ (1 + -^) — an 



g 2 \ ^ g- J ^ ^ 



consequently with dO constant, the ratio varies sensibly as the 

 cotangent of half the angle read. This cotangent expressed in 

 terms of the grade is 



cot. -^ = y \^2 k ^/ {g' + \)-l\ .. . .(12) 



By differentiating the factor in (1) we obtain the ratio 



^= (l T-^.— ^ = fl ±-^.^V...(13) 



m \ g y / \ 99/ 



and from these three we may compute the distance error arising 

 purely from misreading or bad gradviation. If the grade be the 

 argument and is small, i.e., expressed by a large number, 



dm m d 6 , . , , ^ 



= — V 2 kg— 1 approximately. . . .(14) 



may be found a useful formula. 



With the assumed instrument, in which 2i is about 21° 48', and 

 maximum grade 1 in 2h, we find, taking m as 100, the following 

 values for the error, viz. : — 



Angle 30° 60° 90° 120° 150° 



Grade , 40-18 10-72 5-29 3-45 2-71 



Error dmjm . -0013 -0024 -0028 -0026 -0016 



The maximum error is therefore '28 in 100, or say 1 in 350. 

 Error resulting from the imperfect measurement of the angle in (7) 

 will be insensible. 



Probably the most convenient instrument for rough country 

 would be one graded up to 1 in 5. This would be telemetrically 

 a little more accurate than the instrument assumed in the dis- 

 cussion, and with a good telescope ought readily to measure to an 

 average precision of 1 in 500. 



The instrument will fulfil all the functions of an ordinary 

 accurate level, and is, in addition, serviceable as a telemeter, a 

 grade definer, and a rough level, the heights being found by means 

 of the distances, telemetrically determined, and the grades. Its 

 weight is but slightly greater tlian a level, as also its cost. In 

 some form, therefore, it is probable it will be a permanent addition 

 to the instruments of the surveyor and civil engineer. 



