118 Approximate Values of the Radii of Curvature, be. [Aug. 

 i C H x 3-141159, &c. {1 - (| + §^ + ££& + &c.)j 



where d = 1 — ( ^-rrj , CK = semiconjugate and CH = 



semitransverse diameters. — (Mr. Barlow's Mathematical Tables, 

 p. 293.) 



12. The difference between the squares of the sines of any 

 two latitudes equally distant from the latitude of 45° are equal 

 to each other. 



Let A, and A + B = D, represent the degrees equally remote 

 from 45°, then will sin. 2 (45° 4- D) — sin. 2 (45° 4- A) = sin.* 

 (45° - A) - sin. 2 (45° - D). 



Per trig. sin. (45° + D) = (sin. D 4- cos. D) sin. 45°, ther. 

 sin. 2 (45° + D) = (sin. 2 D + cos. 2 D + 2 sin. D cos. D) sin. 4 

 45° = (1 + 2 sin. D cos. D) sin. 2 45° = i + sin. D cos. D 

 (radius unity). In like manner, sin. 9 (45° 4- A) = 4- + sm - A 

 cos. A; ther. sin. 2 (45° 4- D) — sin. 2 (45° + A) = sin. D cos. 

 D — sin. A cos. A. 



We also have sin. (45° — A) = (cos. A — sin. A) sin. 45°, 

 therefore sin 2 (45° — A) = (sin. 2 A + cos. 2 A — 2 sin. A cos. 

 A) sin. 2 45° = (1 — 2 sin. A cos. A) sin. 2 45° = 4- — sin. A cos. 

 A. In like .manner, we have sin. 2 (45° — D) = 4- — sin. D 

 cos. d ; therefore sin. 2 (45° — A) — sin. 2 (45 — D) = sin. d 

 cos. D — sin. A cos. A; which shows that sin. 2 (46° + D) — 

 sin. 2 (45° + A) = sin.* (45° — A) - sin.* (45° - D). 



13. To find in what latitude the difference between the lengths 

 of two contiguous meridional degrees is the greatest. 



Let A denote the degrees in the required latitude, and m the, 

 length of a meridional degree at the equator, and etbe compres- 

 sion; then Art. 5, m{\ 4§esin.* (A + 1°)} — m(\ 4 3 e sin.* A) 



— 3 em {sin.* (A 4- 1°) — sin.* A} = max. Or sin.* (A 4 1°) 



— vsin.* A = max. Per trig, we have sin. (A 4- 1°) = sin. A 

 cos. 1° 4- cos. A sin. 1° s= sin. A 4 cos. A sin. 1°, the cos. 1 Q 

 being very near unity or radius, hence we have sin.* (A 4 1°) 

 = sin.* A 4- 2 sin. A cos. A sin. 1° 4 cos.* A sin.* 1° = sin.» 

 A 4- 2 sin. A cos. A sin. 1°, cos.* A sin. 4 1° being extremely 

 small ; therefore sin.* (A 4- 1°) — sin.* A = 2 sin. A cos. A 

 sin. 1° = max. Or sin. A x cos. A = max. which is well 

 known to be the case when A = 45 degrees, the required 

 latitude. 



14. The differences between the lengths of any two contiguous 

 meridional degrees equidistant from the latitude of 45° are equal 

 to each other. This will readily appear by comparing Art. 12 

 and 13 together. 



15. The same properties that have been demonstrated (in Art. 

 13 and 14), relative to the meridional degrees, may likewise be 

 shown to belong to the degrees perpendicular to the meridian. 



16. If M v , M represent the lengths of any two contiguous 

 meridional degrees, and P', P the lengths of two corresponding 



