- sin DB: sin DC :: cosb: cos (c + Dc); sin Desin DB:: sin c. cos (c + Dc) : sin Db: sin Dc :: cos B: cos (C + DC); d2c= cos B cos C cos B cos (C+DC) - sin DB : sin DC :: sin (B+DB) sin (C + §DC) ' cot B COS C cot C d, C= sinDcsin DB :: d. B= sin (c + Dc) sin 2 B 2 cot c [3] Let b, c, be invariable; then tan DB: tan DC :: tan (B+1DB): tan (C + DC); - sin Da : tanDB:: sin (a+Da): cot (C + DC); sin DA: sin Da :: sin (a+Da): sin b. sin c. sin (4+DA); ག - sin DA: sin DC :: sin (a + Da): cos B; dA= [4] Let B, C, be invariable, then cos B sin a tan Db : tan1⁄2De : tan (b+Db): tan (c+Dc); d2c= sinDA: tanDb: sin (4+DA): cot (c+ Dc); d24= sin≥DA : sin †Da : sin (a+Da) : d4sin B. sin c. sin (4+1DA) tan c tan b sin A cot c sin B.sin C { sin (A + 1DA) } 2 sin (a + Da). sin c sin Da: tan Db : : sin A. sin B sin B.cos c sin a.tan (c+De) If C=90°, then : sin DA sin Dc :: sin (4 + DA) cos A : cos b sin (b + Db)' sin Da: sin Dc :: sin (a + Da). cos a: sin (c + Dc) cos c; If Da is small, these terms will be sufficient; the following series are however still more convergent: Dx 12x+Dx (20+ Da) Dx ;)2 + &c. }. a =10000, and Two terms of the first series, and the first term of the second, will in most cases be sufficient: if x= Da = 1, the common logarithms will be exact to fifteen decimal places. <-1 D ̧log,”1w=log. ̄`v {DTM (Dx)2, (Dx)3 + 1.2 + 1.2.3 (Cagn. Trig. 372-91; L. C. D. 889, 90.) (15.) The sines of arcs for intervals of a degree having been found by preceding methods, (Trig. 42-5.) the sines for minutes may be more easily found by differences: any two differences having been found, the others may be determined by the equation Dr sin x = - (2 sin Dx)2. {D-1 sin x + D-2 sin x}. (16.) The series of differences for any given intervals having been found, the differences for smaller intervals may be found by the following general method: let a be the larger interval, y the smaller, and let x=py, then |