Adams. — On Natural Sines. 413 



A 2 sin x — — 2 2 sin (x + 26) sin 2 # with advantage in the cal- 

 culation of a table of natural sines. 



It appeared that this relation between the sines and their 

 second differences might be a general one, and upon investiga- 

 tion this was found to be the case, the general relation 



A m sin x = - 2 2 A W " 2 sin (x + 28) • sin 2 (A) 



being obtained. 



This will now be proved ; thus we have 



A m sin x = 2 m sin [x + m (y + #)] sin m 

 and similarly 

 A>»-2 sin (£ + 20) = 2'»-2 s in[a;+20+(m-2) fJL + $\ 1 8 i n m-2^ 



c\m— 2 



q to— 2 



sin [a + m (y + #) _ „.] sin m_2 

 sin[ar + m (~ + #)] sin" l ~ 2 



hence a" 1 sin x = - 2 A™ sin (x + 25) . sin^ 



It will now be shown how it is possible by means of this 

 relation to calculate readily the leading differences, and thus 

 dispense with the cumbersome series for these differences 

 given above by Callet. 



For convenience (A) is preferably written 



A m sin x = - 2 (1 - cos 26) A™ -2 sin (x + 26) 



or A sin x = - 2 (1 — cos Aa?) A sin (x + Aa?) 



or A w sin x = — k • A m sin (x + Ax) (A x ) 



where k = 2 (1 — cos Ax) and is a constant depending on the 

 tabular interval only. 



■»-r . TO • 7 f .111 — 2, - . . TO — 1 . ... 



Now, A sin x =. — « • I A sin a; + A sin an (A 2 ) 



hence any difference is expressed in terms of the two preced- 

 ing differences. 



The formation of the leading differences then reduces to 

 the very simple operation shown in (A 2 ) above. It will only 

 be necessary to compare this with the series given above 

 (AQ, A 2 Q, &c.) to see how much simpler the method here 

 described is. 



In Part II. the application of this method to the calcula- 

 tion of a table of natural sines will be given. 



