GEODESY. xlix 



This agrees as it should with the length given by (4) when 2<f> = 90" and A<^ 

 = 90^* 



b. Arcs of parallel. 



The radius of any parallel of latitude is equal to the product of the radius of 

 curvature of the normal section for the same latitude by the cosine of that lati- 

 tude. That is, see Fig. i, r being the radius of the parallel — 



r = p„ cos <^, 



and the entire length of the parallel is — 



2 TT r = 2 TT p„ cos 4>. 



Designate the portion of a parallel lying between meridians whose longitudes 



are Aj and X., by Ai", and call the difference of longitude A2 — Aj, AA. 



Then — 



Arc of parallel AF in feet. 



2 IT p„ COS cf> 



AI^=— 7 AA (in seconds), 



1296000 ^ ^ 



2 TT p„ cos <^ ,. . , . . 



— — He^ — ^^ ^^^ minutes), (i) 



2 TT p„ COS ^ . ^ .. , . 



_ f^n V ^^ / j^ degrees). 



360 V o / 



log (2 77/1296000) = 4.6855749 — 10, 



log (2 7r/2i6oo) = 6.4637261 — 10, 



log (2 77/360) = 8.2418774 — 10. 



A.,, A,, = end longitudes of arc, A\ = A; — A.,, 

 pn= radius of curvature of normal section for latitude of parallel ; for log pn see Table ir. 



Numerical Example. — Suppose <^ = 35°, and AA = 72°. Then from the third 



of (9) 



log. 



cons't 8.2418774 — 10 



Table 11, p„ 7.32 117 16 



cos <^ 9-9133645 — 10 

 ^^ 1-8573325 

 A/' = 2 1 564 827 feet, AP 7.3337460 



* The best formula for computing the entire length of a meridian curve is this : 



in which a, b, and n are the same as defined in section 2. For the values here adopted — 



log. 

 (i + J «■-+•• •) 0.0000003 

 (a 4- b) 7.6209S07 



IT 0.4971499 



length 8.1181309 



The length of the perimeter of the generating ellipse, or the meridian circumference of the 

 earth, is, therefore — 



131 259 550 feet = 24 859.76 miles. 



