394 _ Royal Society :-— 
same eq. as in the preceding; and since its eq. is of the third de- 
gree, it has always one real value. 
Next, let the second surface be a sphere, and you find at least one 
diameter of the first surface perpendicular to its conjugate plane. 
Make this diameter the axis of a, and take for the axes of y and z 
the two principal diameters of the section in the conjugate plane. 
Then D=0, E=0,; F=0; so that the general eq. is reduced to 
Az2?+By?+C2+G=0. Moreover, the system of axes is now rect- 
angular: hence the axis of y, and that of 2, equally with that of 2, 
are each perpendicular to its conjugate plane, and the eq. for p must 
have three real roots, corresponding to these three axes. : 
We might similarly investigate ‘“ the conditions of contact for two 
concentric surfaces ;”’ which, when one of them is a sphere, gives the 
cubic whose roots are a’, 4°, ¢*, principal axes of an Ellipsoid. 
Problem. To discuss the results of Tangential Co-ordinates. 
[This expression is employed as by Dr. James Booth in an original 
tract on the subject. | 
Put P=Azr+Dy+Ez+ A, R=Evr+Fy+Cz+C, 
Q=De+ By+ Fz+B, S=A,r7+By+C~+G | 
Then Pr+ Qy+Rz+S=0 is the eq. to the surface, and Pa! +Qy/! 
4+Re'+S=0 is the eq. to the tangent plane at (ayz). Hence if 
a'y'z! are the three tangential co-ordinates (or intercepts cut from the 
co-ordinate axes by the tangent plane) we have P2!+S=0, Qy'+8=0, 
R-'+S=0. Let énz be the reciprocals of 2! y' 2'. Then P+£S=0, 
Q+nS=0, R+ZS=0; and the eq. to the surface becomes éx+ ny 
+fz—1=0. Restore for PQR their equivalents ; then eliminating 
zyzS you get 
D BB, 
EF C.G,.2.|=0; 
ae BO One) 
bay a 
general eq. to the surface, with axes oblique. 
If the last eq. (developed) be represented by 
a? + by? + cb? + 2agb + 2byn + 2ey6 + 2dén + 20eb6 +2 fn +g=0, 
it is not difficult to obtain a system of eqq. in which aéc...fn¢ play 
the same part, as just before did ABC...ayz. Whence again we have, 
a@dea«£ 
dbf by 
Men at A Pa =0; 
a, 6, Cg g—1 
z y e—1,0 
which is the original eq. of the surface under the form of an Eliminant. 
The most arduous problems (as Dr. James Booth has shown) are 
often facilitated by these co-ordinates ; but without Eliminants, the 
eqq. cannot be treated generally and simply. 
The paper likewise contained the application of Eliminants to tan- 
