10 REPORT—1868. 
rT (M+1) 5 
T(@+l)-F(y+l). Perl) eck 
varies from unity at the corners of the triangle up to the very large number 
T(M+1) , , : : 
Ie(Manye at its centre. No simple and easy method of evaluating precise 
irGtys 
numerical results for the probabilities of the different cases is attainable ; butit will 
readily be seen that conclusions of a general nature, not without interest and im- 
portance, may notwithstanding be obtained in any given case. 
Analogous considerations applied to a four-cornered constituency lead to a re- 
presentation of the different cases by the portions into which a tetrahedron is 
divided by planes parallel to its four faces. 
solution being in fact represented by 
On the Division of Elliptic Functions. By W.H. L. Russetx, FR.S. 
This paper was intended to illustrate some of the discoveries of Dr. Weierstrass. 
It was shown that certain series for sin am. w, assumed to be convergent for cer-- 
tain values of («) when sufficiently small, are true when (w) has any value. The 
proof depended on an application of the proposition known as Abel’s Theorem. 
On a construction for the Ninth Cubic Point. 
By Professor H. J. Srernen Surtu, /.R.S. 
On Geometrical Constructions involving imaginary data. 
By Professor H. J. Sreruen Surru, /L.S. 
On a property of the Hessian of a Cubie Surface. 
By Professor H. J. Srrruey Sura, RS. 
On the Successive Involutes to a Circle. By J. J. SytvesTer. 
From the first involute of a circle we may derive a family of parallel curves 
forming the second inyolutes of the circle; from each of these again families, the 
totality of which will form the third involutes, and so on continually. 
The author had been led by circumstances to study the arco-radial or semi- 
intrinsic equation of these curves, and had arrived at certain conclusions concerning 
its form which subsequent investigations have verified: it turns out that the ge- 
neral equation between the are s and radius vector r of the general involute of the 
mth degree will be found by taking F, any rational integer function of x of the th 
degree, and eliminating x between the equations r?=F?+ Gay 
dx 
s= f ak +o 
Tt follows, as the author had surmised, that the general arco-radial equation for 
the involute of the mth order when » exceeds unity, is of the degree (n+1) in r? 
and 2n ins. Of course, in the case of » equal to unity, the degrees sink to 1 in 2 
and lin s. The second involute formed by unwrapping from the cusp of the first 
may be termed the natural second involute, but is not the most simple of the 
family ; this, which is at the normal distance of half the radius externally from the 
one last named, is of the third degree in 7 and the second in s._ It may be derived 
from the curve which a fixed point in a wall at half the length of the radius of a 
wheel from the ground marks in the wheel as it rolls along the face of the wall 
by doubling the vectorial angles and taking the square out of the radii vectores. 
From the arco-radial it is easy to pass to the general polar equation to the n-ary 
involute ; the equation between p, the perpendicular from the centre and q the 
polar subtangent, is also very easily obtained, being, in fact, no other than the result 
