MISCELLANEOUS EXAMPLES. 397 



5. A figure made up of a rectangle and an isosceles triangle 



is inscribed in a semicircle : determine its dimensions 

 so that its area may be a maximum. 

 Result. The height of the rectangle must be half the 

 radius of the circle. 



6. Find the cone of least surface, excluding the base, that 



can surround a given sphere. 

 Result. The sine of the semi vertical angle = \/2 1. 



7. Find the cone of least surface, including the base, that 



can surround a given sphere. 



Result. The sine of the semivertical angle = $. 



8. Find the maximum value of cos 6 cos < cos ^ where 



+ (> + = TT. 



n rp f U U . 



9. Iranstorm -= 2 + -T-J by assuming 

 cLx dy 



x = jX + m } y, y = 



10. An equation between three variables contains n arbi- 



trary functions of one of them, and 4?i 2 n 1 arbitrary 

 constants : shew that generally the equation must be 

 differentiated at least &n 2 times in order that the 

 functions and constants may be eliminated. 



11. If Fbe any function of x, y, z, and V the value of V 



when vw is substituted for x, wu for y, and uv for z ; 

 then 



^ [ 



2 



/ * ' v 7 2 T 7 fl I * 



au av dw } 



12. If y = e tx + e~, and z 4- xe~*"* = 0, shew that the general 

 term in the value of y when expanded in a series is 



