1«'20.] the Direct Method of Finite Differences. 285 



By substituting in Problem 2, 



A" (cos. x) = COS. X 5 COS. n to — sin, ?i w tan. x 



— 7« Tcos. (n — 1) w — sin. (« — \) w tan. x] 



+ " "~ Tcos. (« — 2) w — sin. (« — 2) w tan. jr) 

 .... + IJ 



The same remark relative to the smallness of the arc w, may 

 be applied to this example as was in the last, and give 



A" (cos. x) = cos. X ^l — mv tan. .r — n f\ — (n — l)wt2Ln.x\ 



+ "_l!i_Li) (!_(„_ 2) «; tan. x) ....+ ij 



Example 10. — To find the «th increment of tan. x (radius 

 unity). 



Put (p = tan. X and A x = w, a. constant quantity, then 



i^M 'ill. -J-" + 'an. »i ii' 1 + tan. n to cot. x 



, &c. 



tan. X tan. x [\ — tan. n tc tan. a) 1 — tan. n le tan. jr 



♦„_i 1 + tan. (n — 1) u) cot. x (?„_, 1 + tan. (n — 2) to cot. x 



tan. a- 1 — tan. (« — 1) lo tan. jr' tau. a- 1 — tan. (n — 2) jo tan, j;' 



By substituting in Problem 2, 



A» (tan. .t) = tan. X \ \ +'■•-"■"'■' ^"t. x _ ^^ /l -. tan, (n-l). cot._.x 

 ^ ^ 1 — tan. n u) tan, j^ \1 — lan,(n — Ijio tan, j:/ 

 n (n— 1) / I + tan, ( n — 2) tc cot. x\ • 



"*" 2 U - tan. (n-2)ui tan. x) ± ^S 



By considering the arc w incomparably small, the following 

 equation will be nearly true, viz, 



. , ,, , , f 1 + mo cot. X /I + (n — 1) tccot, a\ 



A" (tan, x) = tan, x \- — n ( ■; ^^ ) 



^l — nictaii.x vl — (n — 1) ic tan.x/ 



n (ti-1) / I + (n - 2)«cot. j \ , 



+ 2 ll -(n- 2)totau.a7 ""'^ ^\ 



Example 11. — To find the wth increment of cot. x (radius 

 unity). 



Put (p = cot. X = — — and A a- = w, a constant quantity, thea 



1 1 — tan. n to tan. x 1 



ffl —. s= • (B -i- 



'^" cot. i- 1 + Ian. ft ta rnl. r ' rn — 1 



cot. X I + tan, n to cot. x ' f" — ' ' cot.x 

 1 — tan. (n — 1) tc tan. x „ 

 1 + tan. (n — 1) to cot, x' 



By substituting in Problem 2, 



A" (cot. X) = cot. X S ' " '""• " *" '^°- ^ - n /»-»'"■.("-') "'tan. XX 



i 1 + tan. n to cot. x \1 + tan. (ii- 1) to cot. xj 

 n (n — 1 ) /I — tan. (n — 2) to tan. x\ - > 



2 il + tan. [n- 2) to cot. a/ — J 



By considering the arc w inconsiderably small, the following 

 equation is nearly true, viz. 



