752 iiEi'OBT — 1889. 



dence when obser\'ed from any point on a semicircle : the ratio of the chord to the 

 radius being of course 2. 



It is easily seen from the geometrical construction that the length of the chord 

 is twice the sine of twice the angle of inclination of the mirror to the axis of the 

 sight. The ratio of the chord to the radius varies from to 2. 



A scale divided into twenty equal parts, with subdivisions, is engraved on the 

 bar. The graduations are numbered by decimals up to the digit 2. To the mirror 

 is attached a metal plate of an ovoid shape with a bevelled edge. This edge is 

 the fiducial line by which the scale is read. When the plate is in a median or 

 symmetrical position, the edge cuts the centre line of the scale at the extreme 

 graduation marked 2. 



The figure of the curved edge of the plate is a polar curve, whose equation is 



r = a + b sm 2 d, 



where a is the distance from the zero graduation to the axis of the mirror, and b is 

 the length of the scale from zero to 2, and d the inclination of the mirror. 

 A model of a family of curves of the equation 



?• = « ± 6 sin 2 6 

 was exhibited to the meeting. 



The ratio of the chord to the radius being known, the plate is set so that its 

 bevelled edge cuts the scale at the graduation corresponding to the ratio. The 

 mirror is then in adjustment. One side of the plate is for arcs less than a semi- 

 circle, the other for arcs greater than a semicircle. 



The same method was suggested by Mr. R. C. May, and was described in the 

 first volume of the ' Proceedings of the Institution of Civil Engineers,' in the year 

 1841. The instrument was a modified box sextant, and a specially prepared set of 

 tables was required for setting the mirror. An arrow weighted with lead was 

 released by a trigger to mark the exact spot on the ground. 



A telescope may be used, but the instrument is not intended to compete with 

 a theodolite for exact work. It is intended for simple and expeditious work by a 

 single observer without assistants, and for those cases where a light, portable instru- 

 ment, needing no tables or calculations, will be appreciated. 



An ordinary five-inch theodolite weighs 12 lbs. without its case. This instru- 

 ment weighs 1 lb. 10 oz., or with telescope, 2 lbs. 



2. On the Comparative Cost of working English and American Railways.^ 

 By B. B. Dorset, M.Am.Soc.C.E. 



The author exhibited tables comparing the working expenses in 1888 on the 

 London and North Western, Great Northern, Midland, Great Western, and Great 

 Eastern railways of England with those on the Pennsylvania railroad and the 

 KnoxvUle Branch of the Louisville and Nashville system, these two last-named 

 roads representing the extreme types of American railway construction — the first 

 being one of the best, and the last one of the most cheaply constructed roads in 

 America. 



It was shown that on the English railways, which have cost from four to six 

 times more to construct than the American roads, the cost for transporting freight 

 is more than double the American cost, whereas it should be less, or why make the 

 Increased outlay for superior construction ? 



The author thinks that the great difference in favour of American railways in 

 the working expenses is owing principally to the following reasons : — 



1st. The trains on the American railways carry much larger loads than those 

 on the English roads. 



2nd. The universal use on the American railways of rolling-stock with bt%ie- 

 trucks, which run with much less friction and wear and tear than the English 

 rolling-stock with its long rigid wheel base. 



' Printed in extenso in the Engineer, vol. Ixviii., p. 275. 



