104 Mr. Adams's Demonstration [Feb^ 



Then d 1/ = w sec.^ x 



— = ro'^ tan. x sec' x 



— = ^ (2 tan."- :r sec.'^ x + sec." x) 





= -^ (tan.3 X sec- x + 2 tan. x seC* a) 



'±j _ ^ (2 tan.* j; sec* x + 11 tan. « x sec'' x + 2sec.®jp> 



8cc » 



Or by writing 1 + tan." x for sec" x, we have 



A (tan. x) = dy + ^ + ^^ + .j-^ + &c = 



+ A X sec" X 



+ A X'^ tan. ,r sec.'^ x 



+ ^(1^-3 tan."-.r)sec.^ x 



-f -7^(2 + 3 tan.\r) tan. :r sec'a: 



4. ii:! (2 -f- 15 tan." a: + 15 tan.4 ^) sec- x 



4. ^ (17 + 60 tan." x + 45 tan." x) tan. x sec.'^ .t 



&.C ••- 



Example 18. — To find the increment of cot.o: (radius unity). 

 Put w = dx = A X, a constant quantity, and ?/ = cot. x. 

 Then by a process similar to that used in example 17, we have 



A (cct. .r) = — A X cosec.^ x 



+ A .1° cot. X cosec* x 



A l"* 



^ (1+3 cot.* .r) cosec- x 



+ -^- (2 -f 3 cot."- .r) cot. X cosec." x 



_ ^(2 + 15 cot.-.r + 15 cot.^i) cosec- r 



15 



4. ^ (17 + 60 cot.^ X + 45 cot.'' x) cot. x cosec *^ 



45 



8cc "■ 



Example 19. — To find the increment of the sec x (radius onitj^^ 

 By example 17, 

 A (sec. x) = A X tan. x sec. x 



+ —^ (1+2 tan.* x) sec. x 



+ ~Y~ (2 + 3 tan.* x) tan. x sec. x 



+ ~{2 + 13 tan.* x + 12 tan." x) sec x" 



