Adams. — On Nattiral Sines. 205 



Tabular interval = Aa; = 1°. Initial value sin 9°. 

 k = 000030, &c. 

 Now, A sin 9° = cos 9° sin 1° - *-. sin 9° 



= 001721, &c. 



Section 9°-18°. 

 sin 9° = + 0-15643, 44650, 40 



A sin 9° = + 0-01721, 37126, 267 



sin 10° = sin 9° + A sin 9° = + 0-17364, 81776, 67 



A 2 sin 9° = - k sin 10° - 0-00005, 28949, 1709 



A sin 10° = A sin 9° + A 2 sin 9° == + 0-01716, 08177, 096 



A 3 sin 9° = - k A sin 10° = - 0-00000, 52273, 51315 

 A 2 sin 10° = - 0-00005, 81222, 6840 



A 4 sin 9° = - k A 2 sin 10" = = + 0-00000, 00177, 04606, 

 d 3 sin 10° = - 0-00000, 52096, 46709 



A 5 sin 9° = + 000000, 00015, 86908, 85 



A 4 sin 10° = + 0-00000, 00192, 91514, 9 



A 6 sin 9° = - 000000, 00000, 05876, 382 



&c. 



These leading differences are sufficient to determine the 

 sines to twelve places of decimals, the value obtained for 

 sin 18° being 0-30901, 69943, 75, which checks the work; and 

 the other values, as taken from the working schedule without 

 any alteration or adjustment, are : — 



Sines. Sines. 



9° 0-15643, 44650, 40 14° 0-24192, 18956, 00 



10° 0-17364,81776,67 15° 0-25881,90451,03 



11° 0-19080, 89953, 77 16° 0-27563, 73558, 17 



12° 0-20791, 16908, 18 17° 0-29237, 17047, 23 



13° 0-22495, 10543, 44 18° 0-30901, 69943, 75 



3. The next example selected is where the tabular interval 

 is 3', and values approximately correct to eleven decimal 

 places are required. 



Tabular interval =/\x = 3'. Initial value 9°. 



sin 3' = 0-00087, 26645, 15235, 14954, From ser.es in 



3304 Part L 



COS 3' = 0-99999, 96192, 28249, 43113, -Prom series in 



77097 Part L 



.-. k = 0-00000, 07615, 43501, 13772, 45806 

 also A sin 9° = cos 9° sin 3' — ~ sin 9° 

 = 0-00086, 18610,"bl 



