MISCELLANEOUS EXAMPLES. 225 



16. Transform the expression 



du\* fdu\ z (du\*} ( du du du}~ 2 



into one in which r, 0, (j> shall be the independent 

 variables, having given 



x = r sin 6 cos <, y = r sin 6 sin <j>, z = r cos 6. 



17. If x, y and f, 17 be co-ordinates of the same point 

 referred to two systems of rectangular co-ordinates, 

 shew that 



^ ^ _ ( d ^ V = f?! ^ _ f ^ V 

 ~ 



18. Shew that x* + x sin x + 4 cos x is a minimum when 



# = 0. 



19. (7$ is the perpendicular from the centre G of an ellipse 



on the tangent at a point P: find the maximum value 

 otPQ. 



Result a b. 



20. A straight line drawn from the extremity of the minor 



axis of an ellipse cuts the major axis at Q and the 

 curve at P; from P the ordinate PN is drawn to the 

 major axis : find when the area PQNis a maximum. 



Result. PN= (V17 - 1). 



T. D. C. 



