412 Transactions. — Chemistry and Physics. 



- ^5? (7920m 7 + 110880m 6 Am + &c.) 



+ ll^L (17160m 9 + 308880m 8 Am + &c.) 



_ 1 1-^ (32760m 11 + &c.) + &c. (A 4 Q) 



. , • i» i 120c ■ Am 5 d- Am 5 /cie . nr . „ , inmA . 



A° sin — g- = - 6 — — — (2520m 2 -f 12600mAm + 



n l n " 16800AW 8 ) 



+ il^L (15120m 4 + 151200m 3 Am + 604800m 2 Am 2 

 + &c.) 



_ L-^. (55440m 6 + 83 1600m 5 Am + 5544000m 4 Am 2 

 + &c.) 



+ 0_^1 5 (154440m 8 + 3088800m 7 Am + &c.) 



h ■ Am* 



a' 



(360360m 10 + &c.) + &c. (A 5 Q) 



A ,; sin ^y = - ^-^~ (5040m + 15120Am) 



+ ^— ^ (60480m 3 + 544320m 2 Am + 

 1723680mAm 2 + 1905120Am 3 ) 

 _ frtefi ^gg 2640m 5 + 4989600m 4 Am + 

 " 3 1600800m 3 Am 2 + &c.) 



+ ? ' ^» 6 (1235520m 7 + 25945920m 6 Am + 



n ' &c.) 

 _ h -*f (3603600m 9 + &c.) + &c. (A 6 Q) 



Ainsi des autres. Nota. — Ces expressions Am 2 , Am 8 , &c, 

 tiennent lieu de celles-ci (Am) 2 , (Am) 3 , &c." (Callet, pages 

 59 and 60.) 



Other expressions for the differences of sin x are, if Ax = 

 20, as shown below, — 



A 4w sin x = 2 4? ' sin {x + toiO) sin 4,, 



A 4?i+1 sin x = 2 4,i+1 cos (x + 4^T~1 9) sin 4 "* 1 * 



A 4,i+2 sin x = - 2 4 " +2 sin (x + 4^T2 0) sin 4n+2 



A sin x = — 2 cos (x + 4?i + 3 0) sin U 



(De Morgan, " Differential and Integral Calculus.") 



These four expressions may be combined into one expres- 

 sion as follows (Boole, "Finite Differences") : — 



A m sin x = 2 m sin \ x + m (^ -f 6 J sin w 



Now, Colenso ("Plane Trigonometry") shows that when 

 the tabular interval is small it is possible to use the expression 



