TRANSACTIONS OF THE SECTIONS. 



13 



requires that ( - V be > ~. In particular, examine 



^y=(.i--3)-^-x-i = (3-^)^-.i-i. 



You find /i'-y- positive while a-=y, and still positive when .r=2^; but negative 

 when :r=2}j, or when .r=|. Thus the interior oval has limits somewhat short of 

 x = }, .v= 2h ; while the height of the cloister is 3. 



On the Curvature of Surfaces of the Second Degree. By F. "W". Newman. 



Writing for the equation 



A.r-+B/ + C3H2Dxy+2Exz + 2Fy2+25d.r+2/3y+2y2+G=0; 



taking also 



V= 



and W= 



we know that in ellipsoids and hyperboloids both V and W are finite ; in the Cone 



V is finite and W=0 ; in Paraboloids V is zero and W finite ; in Cylinders both 



V and W are zero. 



This paper shows by a direct process, (1) that if the axes move parallel to them- 

 selves without change of A B C, W remains unchanged ; (2) that if the axis turn 

 about a fixed origin while C remains unchanged, the change in tlie value of W is 

 easy to estimate. In fact if W; be the value when the axes are rectamjnlar and W 

 belong to the same surface estimated from another system of axes in which 



Do=co3 (.r, y ), Eo=cos (x, z), Fo=co3 (y, z), 

 it is here proved that 



1 D„E„ 

 W=W: D„ 1 F„ 



EoF, 1 



If then we take lengths 0jr=0y=0j=:l along the axes, W is constant, while the 

 volume of the parallelepipedon, whose edges are Ox, Oy, 0:, is constant. 



The relation of the function W to the curvature is cardinal ; W cannot chano-e 

 sign in a given sm-face ; and when W is positive, the curvature is concavo-convex. 



On Conic Osculation. By F. W. Newmait. 

 The topic was treated from the general equation 



Aa;=+ByH C + 2Diry -h 2Ea;+ 2Fy =^0. 



First. Taking the origin on the curve, C=0. It then is sho-nm that two curves 

 which have a common tangent at a common point may be denoted by 



J A iB2+ByH2Diry+2E x=Q, 



and consequently that the two osculate if Eo=:E. 

 It immediately follows that (if 8 is the obliquity of the axes) 



B(a;-+y2+2xy cos S)-|-2Ex=0 



is the osculating circle. From this the results of the common treatises are deduced 

 very simply. 

 Secondly, it the origin be still In the curve, but neither axis be a tangent, put 



V- 



