ELISHA MITCHELL SCIENTIFIC SOCIETY. II7 



telescope and set to a^-^ = 38^, a^.^ = 26', ^^^3 = A3 = 18' 

 successively to fix stas. (2), (i) and (o). 



Lastly, at sta. o or S, with last vernier reading ^3 = 18' 

 on plate, sight to sta. (3) and turn to o°oo' to sight along 

 tangent SY. 



Always set the angle off, that any chord makes with the 

 Y axis, o?i the proper side of the o pointy so that w^hen we 

 sight along that chord and then turn to o°oo' the line of 

 sight wnll be parallel to the tangent SY. This is best done 

 by leaving the last angle turned clamped on plate, when 

 we move up to a new station, at which point verify angle 

 and reverse telescope to sight back to last station at which 

 transit was set. 



APPROXIMATE FORMULAS. 



By referring to the general table, we see that the ordi- 

 nate X at the middle of any length of curve is nearly yi 

 that at the end. Thus the ordinate Xg = 12.56 and y^ 

 X12 = 12.51, also X3 = 1.57, and this is equal yi Xg = 

 1.57, etc. 



If we use the approximate formula x = ^ a^, found 

 from (14) by neglecting all terms after the first and desig- 



s 

 nate by .r,, the ordinate corresponding to s^=z — we have 



2 



-^o = /-B I — I or \i the extreme ordinate x. 



■'(t)°" 



The equation x = V^ ay^ is that of the cubic parabola, 

 and we see that the eq. x =z j^ as^ closely approximates to 

 it for the very flat arcs considered, thus furnishing the basis 

 for the approximate solutions before referred to. 



Again in fig. i, for flat arcs, we have seen that radius 

 OD produced, drawn ± SK, nearly exactly bisects the curve 

 SEL, hence SG is nearly equal to LD (where G is the in- 



