Solution of a Mathematical Problem. 51 
- 
If in equation (6) we take the lower sign and make 7’= yz we 
shall get 
y= ir cos. 6+ 3rcos. 34=rcos. 30 ) (7); 
e=irsin. §+3rsin, 36=r7sin.36 § , 
hence from ( 7) y? =r? cos.20 
gta 
consequently by adding, we get yi tot mrs, which is the same 
as equation (5). Also in Leroy’s Descriptive Geometry this is 
proved to be the equation of the hypocycloid. 
If we differentiate equation (7) we shall get 
dy = — 3rcos.*4sin. 6d0 : 
: eel oe 
dz = -+-3r sin.*6 cos. 644 
Get Bs ; 
hence {(dy? +dz? )? =8rfsin. 6 cos. 0d0 = gr sin. ? 6= length ° 
the curve: taking the integral from 6=0 to 6=90°, it becomes 
3 
=57r. From equations (7) and (8) we get 
Sydz=3r*fsin.*9cos,46d6: taking the integral | 
‘ 4 r2 
from 9=0 to 6=90°, this becomes “sr for the area of the curve 
between the axes cry. : er 
Cor. 1. Equation (5) shows that our problem includes question 
8, No. 11, Leybourn’s Math. Repository, which reads as follows: 
If from one of the angles of a rectangle a perpendicular be drawn 
to its diagonal, and from their intersection lines be drawn per- 
pendicular to the sides containing the opposite angle, then put- 
ting (P,p) for the last perpendiculars, and (D) for the poe Nai 
Cor. 2. Equation (5) is identical with the result obtained for 
a question I proposed in the Mathematical Miscellany for —_ 
in which it was required to find the locus of the points so situate 
within a right angle, that the straight line whose length is (r) 
- terminates in the sides of the right angle, shall be a minimum 
or each of the points. ery 
Cor. 3. acim (5) furnishes a solution to No, 12 a oo 
Cambridge Problems for 1803, which requires the length of - 
longest ladder that can be slided up a perpendicular wall ing 
horizontal plane, under an obstacle given in position—ry ses f 
‘given as co-ordinate of the obstacle, aod (7) aga be: the tength 0 
the ladder; hence we have r= y? +2*)? the length required. 
In Wright’s solutions of the Gmabridee Problems, this question 
is erroneously solved; his result will give r=(y+2)"* 
