I 



I 



On Maxima and Minima of Functions^ Sfc. 89 



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



X 



Aa-a^+ani.!. .^4- ^Jz^jlj (x^-a^) ~^ 



nun. 



which gives the following equation, expressing the relation 

 between x and a: --^— =3a^ . h.l orhv 



^ .. o„...t., 2/ + ^«^+y 



3 



restoring yj ^^ + a^l^ — a^=3a27/ h,l. 



a 



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- 



y+ v/y2+a^ 



meter = 2a) =='V^2+rt2 j^a\\X ^ , and u 



5 or varies as y^. Hence; as — = max., or -- == min., 



^ 



Za 



v' ' Xih 



2 



tf~2V y^ -^a^ +tt y ^* "• h = njjn. whence 



y V y^ + a'' ==3a^\uL^~ — i— I . 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: ^^-^- = 3h.l.cot, 



' ^ * sin. <^ 



^*P- Or if COT. denote a tabular artificial cotangent, and m 



cot. o 3 



the modulus of the common system, ^.^^' ==- (cot. ^9-10). 

 Cor. Since z, (the length of the curve,) =r-^^2 4.^24. 



3 



ah.h y + ^y'+"', hy substitution, z = j (^+3/^ ' 



^4x-+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 f that of the tangent- 



Vol. v.. ..No. J. 12 



