204 Transactions. — Miscellaneous. 



2. As the tabular interval becomes smaller the value of k is 

 reduced, with a corresponding reduction in the labour of 

 forming the leading differences. Thus in the formation of 

 a table of sines for every degree we have, from the series 

 in Part I., — 



sin 1° = 0-01745, 24064, 37283, 51281, 94189, 78 

 cos 1° = 0-99984, 76951, 56391, 23915, 70115, 59 



The squares of these are : — 



sin 2 1° = 0-00030, 45864, 90452, 13499, 68782, 7997 

 cos' 2 1° = 0-99969, 54135, 09547, 86500, 31217, 20 



hence sin 1° and cos 1° are correct,* and k = 2 (1 — cos 1°) 

 = 000030, &c. 



The working schedule for the formation of the leading 

 differences is accordingly as follows : — 



Tabular interval = A# = 1° 



k = 0-00030, 46096, 87217, 

 52168, 59768, 818 

 sin 0° = 0-00000, 00000, 00 



A sin C = + 0-01745, 24064, 373 



sin 1° = sin 0° + A sin 0° = + 0-01 745, 24064, 37 



A 2 sin 0° = - k . sin 1° - 000000, 53161, 721 



A sin 1° = A sin 0° + A 2 sin 0° + 0-01744, 70902, 652 



A 3 sin 0° = - k. A sin 1° - 0-00000, 53145, 5271 



A 2 sin 1° == A 2 sin 0° + A 3 sin 0° = - 0-00001, 06307, 248 



A 4 sin 0° = - k. A 2 sin 1° = + 0-00000, 00032, 38222 

 A 8 sin 1° = A :! sin 0° + A 4 sin 0° = - 0-00000, 53113, 1449 

 A 5 sin 0° -- - k. A :; sin 1° + 000000, 00016, 17S77, 8 



&e. 



With these leading differences the table may be constructed 

 up to 60°, or to 90° if preferred; however, unless the work 

 is carried out to considerably more decimal places than are * 

 here shown the final figures of the higher values will not be 

 accurate. 



It seems preferable, therefore, to construct the table in 

 sections, 0°-9°, 9°-18°, 18°-27°, &c. 



The values of sin 9 . sin 18°, &c, are obtained from 1 above, 

 and the formation of the table for the section 9°-18° will 

 then proceed as follows : — 



* See the value of sin 1° in Logarithmic Tables, R. Shortrede — 

 sin r = 0-01745, 240G4, 17275, M 

 which is 0-00000, 00000, 20007, 97 in error. 



