A S Y 



H. In the hyperbola the area- between the asymptotes fl. -?"'*.' '' ' 



.'assuming yx w), tho hyperbola being oqui-lateral ; .'. area nit 

 log. x -f- C ; and assuming it o when x w, wo shall have in- log. 



as the general expression for the area. 



Ifm rz 1, the areas are the hyperbolic logarithms of the corresponding 

 abscissae ; and hence the origin of the term hyperbolic as applied to lo 

 garithms. 



For Areas of Spirals. See Spiral. 



ARITHMETICAL Progression. See Progression. 



ASSURANCE on Lives. See Annuities. 



ASYMPTOTES, to draw. 



Find the value of ~ = subtan- 



dy 

 gent M T ; .'. A T = -^- x is 



known. Now suppose x to become 



infinite, and T to move on to C ; then _ 



if A C be finite the curve admits an ^ T A M L 



asymptote. Next find the ratio of T M : M P, which, if we again sup- 

 pose x infinite, gives us the ratio of C L : L x ; then by similar As C L : 

 L.r :: C A : A R, of which proportion the three first terms are known, 

 and .". A R can be determined. Join C R, and produce it indefinitely, 

 and C R is the asymptote required. 

 Ex. 1 .To drfcw an asymptote to the common hyperbola. 

 + A>2 



* (when x is infinite) = a = A C. Again 



T M C M P :: 2g " r + ;g * : 1 V-2 a x + x* ;: ( when x is infinite) r : 

 a + x a 



~ I : C L : L x : : C A (a) : A R, .'. A R b ; Hence from A draw 



A R = b ; take C the centre, and join C R, and produce it indefinitely, 

 and C R x is the asymptote. 

 EA-. 2. Let the equation be y 3 a x* -f A- 3 . 



Proceed just as before, and we get C L ~ r, L -r < x i A C , ,% 



