On Maxima and Minima of Functions^ ^c. 89 



and the latter as x^—a\ Hence we have to make 



which gives the following equation, expressing the relation 



, , x^—a^ x-\-J x^ —n^ 

 between x and a: -— ===:3a^ h.l. -— !_.„„u,, 



2/ + \/ a2_j_y3^ 



restoring y; y^ +a^\^ —a^=^a^y\\,\. 

 Prob. X. 



Having given the length of a parabola, it is required to 

 find when the area contained by it and a double ordinate is 

 a maximum. 



Here v (putting the double ordinate =2y, and the para- 



meter = 2a) =-Vy^-\-a^ +ah,l.-- ^ , and u = 



^ , or vanes as i/% Hence, as — = max., or -r- = min. 



1 / -3 y "^ V i/^ +a^ 



^--^Vy^+a^ +a^y a.h.l. ^—L — = min. whence 



y \/'^M^=3a2h.l. 2(±— 1.-±^_. This equation will be 

 more conveniently computed by approximation, if a be assum- 

 ed = l. Or if(? be the arc, whose cot. =^, the value of y may 

 be approximated by means of tables of natural and loga- 

 rithmic sines, from the following equation: ^° ' "^ — Sh 1 cot 



-^ sin, <p - . . 



i(p. Or if COT. denote a tabular artificial cotangent, and m 



the modulus of the common system, ^?-^=- (cot. -i(p-10). 



Cor. Since z, (the length of the curve,) =^-Vy^+a^ + 



ah.], y'^^y "^" , by substitution, z == | C~ -\-y^^^ — a. 



^4x- A~y^, (x being the absciss,) when, with a given length, 

 it contains the greatest area possible. The subtangent 

 =2x; hence the length of the half of the curve, which lies 

 above the axis, is | that of the tangent. 

 Vol. V....NQ. J, J2 



