286 Mr. Adams on [OcT. 



A- (cot. X) = cot. X SLZU112^ _ „ /IzJn-^lh^i^n^] 

 ^ ^1+nuJCot. .T M + (n— l)u>cot. ar/ 

 n(n - 1) /I - (fi — 2) lefan. x\ » 



"^ 2 il + (n - 2) 10 cot. .r/ ± ^ J 



Example 12. — To find the ?/th increment of sec. x (radius 

 unity). 



Put <p = sec. X and A x = w, sl constant quantity, then 



(p„ sec. X sec. n w 1 



sec. j: sec. x (1 -' tan. x tan. n ic) cos. n ao — sin. n lo tan. x ' 



^n-l J _^ <Pn-t 



sec. j: """ COS. (n — 1) lu — sin. (n — 1) to tan. x ' sec, x 



. L &c. 



COS. (n — 2} 10 — sin. (n — 2) tc tan. x' 



By substituting in Problem 2, 

 A" (sec. a:) = sec. j; J : 



^ ( COS. n 10 — sin. nu) tan. jr 



+ 



COS. (n — 1) 10 — sin. (n — I) lotan. x 

 n (n - 1) 



2 ( COS. (n — 2) 10 — sin. (n — 2) 10 tan. X ) C 



By considering the arc w incomparably small, the following 

 equation will be nearly true. 



f 1 n 



A" (sec. x) = sec. a: i- 



^ <1— nictan. X 1 — (n— 1) lo tan. a: 



^ n(„- 1) 



;l ^" - '^ . . . . + 1 > 



(l — (n — 2) ictan. x) C 



Example 13. — To find the «th increment of cosec. x (radius 

 unity). 



Put (p = cosec. X and A x = %v, a constant quantity, then 



(f" cosec. X cosec. n 10 cosec. n to 



cosec. a: cosec. x(cot. x + cot. nx) cot. x + cot. nto 



1 



cos. n 10 + sin. n to cot. x- 



tn— I 1 'Pu — 1 



eosec. X cos. (n — 1) to + sin. (n — l)io cot. x ' cosec. x 



\ Sec. 



cos. (n — 2) to + sill, (n — 2) to cot. x' 



By substituting in Pi-oblem 2, 

 A" (cosec. x) = cosec. x \ • 



(COS. n to + sin. n to cot. x 



cos. (n — 1) to + sin. (n — 1) to cot. x 



2 I cos. (n — 2) to + sin. (n — 2) locot. x 1 \ 



By considering the arc w incomparably small, the following 

 equation will be nearly true. 



