614 Proceedings of ihe Poycd Society 
the quantity under the radical sign becomes a square, and in this 
case 
d l — {45. cos 6 2 q- 2 5 } d 0 
= 2 5 { cos 2 6 + 2 } d 6 , 
whence, on integrating, we at once obtain 
l — b { sin ? 6 4- 4 6 } = y 4 5 6 . 
The expression for the radius of curvature also takes a very simple 
form, it is 
_ 5 (cos 2 6 + 2) 2 
\/2 sin 6 
No other curve of this class, nor indeed any belonging to the more 
general formula 
x = a . sin (p 0 + k) , y — b. sin (j 0) , 
seems to be susceptible of easy rectification. 
These results may be exhibited geometrically thus:—Having 
drawn 0x4, OB in the directions of the length and breadth of the 
curve, and described round 0 a circle with the radius OB = OC 
= 5, let OA be made equal to four times CB, and an hour-glass 
curve be constructed in the usual manner. Then, having as¬ 
sumed any arc CD to represent 5. 2 6 and drawn DFQ parallel to 
OA, if FP be laid off equal to a. sin 6, P is a point in the curve, 
and the length from 0 to P is equal to the sum of OF, and twice 
the arc CD. 
Hence it follows that the portion PQ of the curve, cut off b} r the 
line DQ, is just double of the circular arc DBF, cut off by the same 
line. 
B 
Hence it appears that the length of the quadrant OPQA of the 
curve is just equal to the circumference of the circle, or that the 
whole curve is equal in length to four times the circumference of 
the circle described with the radius OB. 
