MISCELLANEOUS EXAMPLES. 419 



r* sin = a* cos 20, 



15. $ and Zf are two fixed points, and a curve is described 

 such that, if P be any point in it the rectangle con- 

 tained by SP and HP is constant: shew that the 

 straight lines drawn from 8 at right angles to SP and 

 from H at right angles to HP meet the tangent at P at 

 points equidistant from P. 



fir ?y \ zx* tj 



10. If / j - , T j be a rational homogeneous function of - , *^ 



of n dimensions, shew that the envelop of the curves 



/Ct/ li\ 



represented by the equation / ( - , f J = 1, under the 



\CL OJ 



condition ab = constant, consists in general of n rect- 

 angular hyperbolae having the axes as asymptotes. 



17. If any quadrilateral ABCD change its form, its sides 



remaining constant, shew that the variations of the 

 angles A, B, C, D are ultimately in the same ratio as 

 the areas of the triangles BCD, CD A, DAB, ABC. 



18. In Art. 274, if p = n 1, we have approximately when 



x and y are very large 



^H-, where 6 = --: 

 x x' $(/iJ 



shew that if q = n 2, we have by continuing the 

 approximation 



y_ b 2 X (/0+2S 



Hence shew that in general the two extremities of 

 the rectilinear asymptote are on opposite sides of the 

 curve. 



