of Edinburgh, Session 1871-72. 
613 
6. On Geometric Mean Distance. By Professor 
Clerk Maxwell. 
7. On a Singular Case of Rectification in Lines of the 
Fourth Order. By Edward Sang, Esq. 
The class of curves resulting from the formula 
x — a . sin 0 , y = b . sin 2 0 
are of considerable interest as occurring in various mechanical in¬ 
quiries. When a straight wire, whose effective breadth and thick¬ 
ness are as two to one, is fixed at one end and made to vibrate, its 
free end describes a curve of which the general equation is 
x = a . sin (6 + k) , y = b . sin 2 6 , 
in which h is constant for the particular variety of curve. When 
T 
h — -^-7r the curve becomes a parabola, and when h — o , it takes 
the form above mentioned ; these phases were exhibited by me in 
1832. Again, when a system of toothed wheels is deduced from a 
straight rack, having a curve of sines for its outline, the points of 
contact describe a curve of this class, as is shown in my treatise on 
the teeth of wheels. 
In attempting the rectification of these curves, we have to inte¬ 
grate an expression of the general form 
dl — | a 2 . cos 0 2 + 4 b 2 . (cos 2 0) 2 yd 0 , 
and for this purpose have to expand the root in an interminate 
series, and then integrate each term, the result being unmanage¬ 
able from its complexity. In one particular phase of the curve, 
however, the integration can be easily effected. The above general 
expression may be written 
dl = [ 16 b 2 . cos 6* -f {a 2 - 16 b 2 ) cos 0 2 + 4 b 2 y d 6 , 
and we readily observe that if a 2 = 32 6 2 , that is, if a = 4 b*f 2 
4 x 
VOL. VII. 
