398 MISCELLANEOUS EXAMPLES. 



13. If = x+a^(y}+l3<f)(y} + ...... , then 



14. If y z -f x$ (y), and y z + x'-fy(y''), z and z being 

 independent variables, shew that the general term in 

 the expansion of f(y, y'} in powers and products of x 

 and x is 



Find the coefficient of x'x' in the expansion of. 

 cos (ay + ofy'}, when y=z+x siny, and y'=z'+x sin ?/'. 



15. In any curve the part of the tangent between the point 



of contact and the perpendicular from the origin on the 



tangent is equal to -7- . 

 as 



16. Shew that the equation to the normal at any point of a 



curve may be put under the form 

 x x _ 7/ >/ 

 d*x d'y 

 ~d? d? 



Shew that this equation is the analytical expression 

 of the fact, that if a tangent be drawn to a curve at 

 any point P, and in the tangent PT be taken equal to 

 the arc PQ and on the same side of P, then the straight 

 line QT is ultimately perpendicular to the tangent. 



17. In the ellipse the focal distance cuts the curve at an 



angle, the tangent of which is a mean proportional be- 

 tween the tangents of the angles at which the corre- 

 sponding diameter and a parallel through the point to 

 the transverse axis cut the curve. 



18. If a curve be referred to axes inclined at an angle a to 



each other, shew that the radius of curvature is 



du fdii 



1 + 2 cos a -f + ( -f- 

 dx \dx 



sma- % 

 ox 



