530 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



zontal absciss, while this ordinate itself is the derivative of the area, wherefore 

 the area of the side elevation is proportional to its second derivative, and is there- 

 fore expressed by a recurring function of the second order ; and so also must be 

 its derivative the vertical ordinate ; hence the outline of the arch must be of the 

 nature of a catenary. 



17. The compounds of these catenarian functions are not less interesting than the 

 functions themselves. Thus, we may form compounds analogous to the secant, 

 cosecant, tangent, and cotangent of the angular calculus, and, for want of other 

 symbols, we may designate these by underlines ; thus we may put 



1.1 . cat t . sus t 



sec t = ; cse t — ; tan t — ; cot t . 



— sus t — cat t — sus t — cat t 



whence we readily find 



tan t 2 + sec t 2 = 1; cat t 2 — cse t 2 = 1, 



and by differentiation 



sec t = — sec t . tan t ; cse t — — cse t . cot t > 



tan t = + sec t 2 ; cot t = — cse t 2 



u — — u — " » 



log sus t = tan t ; log cat t — cot 2 t , 



i — i< — > 



log tan t — tan t + cot t = 2 cse 2t . 



18. By taking the inverse of these functions we obtain another set of differ- 



entials; thus, on putting sus t = x, we have cat t = V(% 2 — 1) and t =sus x, so 



that the equation sus t = cat t becomes 

 it 



-i -i 



sus x = (x 2 — 1) ; similarly 



—1 

 cat x 



\x 



= 



(x 2 + 1) * ; 



—i 

 sec x 



\x 



= 



-1 



as yf (1 - x 2 )' 



—1 



= 



-1 



\x 



* V (1 + as 2 ) > 



tan x 



\x 



= 



(i - tf 2 ) -1 ; 



cot X 

 1* — 



= 



(1 - x 2 ) 1 . 



