-1- BOOK OF MATHEMATICAL PROBLEMS. 



1018. Perpendiculars are let fall upon the tangents to :ui 

 ellipse from a point within it, whose distance from the centre 

 is c : prove that the area of the curve traced out by the foot 

 of these perpendiculars is 



1019. Find the whole length of the arc enveloped by the 

 directrix of an ellipse rolling along a straight line during a com- 

 plete revolution; and prove that the curve will have two cusps if 



/5 1 

 the eccentricity of the ellipse exceed ^ . 



'2 



1020. Two catenaries touch each other at the vertex, and the 

 linear dimensions of the outer are twice those of the inner; two 

 common ordinates MPQ, mpq are drawn from the directrix of the 

 outer: prove that the volume generated by the revolution of Pp 

 about the directrix is 2v x area MQqm. 



1021. Find the limiting values of 



f . ... 27T . 3?r _.irVi 



(1) -{sin sm sin ... smm !)-> , 

 ( n n n ' n) 



/n\ f Tf ' 9 2?T . a 3?T . TT) 



(2) <sm- sm 2 -sin 8 ... sin"" 1 (n-l}-\ , 

 I n n n ' n) 



when n is indefinitely increased. 



1022. If P m = f *V( 2 - 3x + x 2 ) dx ; then will 



*\ 



2 (w + 2)P m - 3 (2m + 1) P mM + 4 (m- 1) P m _ 2 == 0. 



p 



1023. An arithmetical, a geometrical, and a harmonical pro- 

 gression have each the same number of terms, and the same first 

 and last terms a and I ; the sums of their terms are respectively 



