GEODESY. lix 



.a„ ;(u" + AX + = J-°^«9~^^^4^ cot I a' 



COS ^,(90 — ^1 -j- 7/) 2 



tan K«" - A,V + {) = t f^ ~ t' I "> <=<>' ^ «■ 



Sin i(9o — </)i + 7/) 



Pm sin ^(a -|- a -|- g *■ 11-' -V -'i 



(4) 



(5) 



To express rj, I, and ^o — </>i i" seconds of arc we must multiply the right hand 

 sides of (2), (3), and (5) by p" = 206 264."8. For logarithmic compution of 7/" 

 and c,", or ?/ and t, in seconds, we may write with an accuracy generally sufficient 



log 7/' = log (p"Vp«) + h -^2 (" ) ' COS^ c^i COS^ a', (6) 



^2 



log i" = log 1 ^-^:r72j^' + log {(v'7 cos^ «^, sin 2 a'}, (7) 



where fx in (6) is the modulus of common logarithms. For units of the 7th deci- 

 mal place of log rj" we have for the adopted spheroid 



^°Si^-372=3-69309- 



Also 



e^ 



^°g^ (x-.V ^^-9^^97-10. 



Similarly, for the computation of the logarithm of the last factor in (5) we have 



log {i + tV v' cos^ Kaf' - a')} = log {i + -±^^ (:n"r cos^ K«" - «')}• 



Putting for brevity 



!?=7::7^2(VTcosn(a"-a') 



the logarithm of the desired logarithm is given to terms of the second order 

 inclusive in q by 



log log (i + ^) = log /x ^ - I /x q. 



For the adopted spheroid 



/A 



log —7^7X0= 4-92975 - 10 

 I2(p )- 



for units of the seventh decimal place. 



For a line 200 miles (about 320 kilometres) long, the maximum value of the 

 second term in (6) is but 12.6 units in the 7th place of log?/'. For the same 

 length of line, the maximum value of C' is o."895, and the maximum value of the 

 logarithm of the last factor in (5), or log (r + q), is less than 922 units in the 

 seventh place of decimals. 



For computing differences of latitude, longitude, and azimuth in primary 

 triangulation whose sides are 1° (about 70 miles, or 100 kilometres) or less 

 in length, the most convenient means are formulas giving <^2 — </>i> ^^2 — •^i) and 



