ELISHA MITCHELL SCIENTIFIC SOCIETY. 107 



since a ^ o when s = o. Also from (i) r = 00 when s =^ o. 

 From the differential triangle we have, 



dx = ds sin a = sin (^as^). ds . , . , (3), 



dy = ds cos a = cos (as^). ds (4); 



but unfortunately we are not able to integrate these 

 expressions in finite terms, so that the equation of the curve 

 in terms of x and y cannot be obtained. 



In practice the curve SL will be run by measuring X 

 chords of c feet each along the curve. When these chords 

 are sufficienth' short they can be regarded as of the same 

 length as the arcs subtended by them, hence we shall 

 alwavs write, 



^ = Nr (5), 



for the length of curve from S to any tangent point con- 

 sidered as L. 



The degree of curve at L, D° being equal to the radius 

 of a 1° curve divided by r, we have, since the radius of a 



(as the sines of small an- 



si'n {y,f 



\l^o) 



gles can be taken as equal to the arcs themselves), 



18000 36000 

 D° = = as (6). 



To express the arc a (eq. 2) in minutes, we notice that 

 its ratio to a semi-circumference whose radius is one, is 

 a 



— and multiplying this by 180 X 60 we find a in min- 

 utes. Hence from (2) and (5), 



a 180 X 60 

 a (in minutes) = — 180 X 60 = ^ c^ N^ 



