431 



x'y'z' are the three tangential co-ordinates (or intercepts cut from the 

 co-ordinate axes by the tangent plane) we have Px' + S=0, Qy' + S=0, 

 Rs-'+S=0. Let 17 z be the reciprocals of x' y' z'. Then P+S=0, 

 Q + jjS=0, R+S=0; and the eq. to the surface becomes fcr-f-ijy 

 + ^1 = 0. Restore for PQR their equivalents ; then eliminating 

 xyzS you get 



A D E A 2 I 



D B F B 2 r, 



E F C C 2 C =0; 



A a B 2 C 2 G-l 



r, C-l 



general eq. to the surface, with axes oblique. 

 If the last eq. (developed) be represented by 



it is not difficult to obtain a system of eqq. in which abc...^ play 

 the same part, as just before did A.BC...xyz. "Whence again we have 



a d e a 2 a? 



d b f b, y 



e f c c 2 z 



, b, c, g-l 



x y zl 



which is the original eq. of the surface under the form of an Elimi- 

 nant. 



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- 

 gential co-ordinates in Curves of the Second Degree ; and urged that 

 eliminants ought to be introduced into the general treatment of these 

 curves also, if only in order to accustom the learner to their use and 

 gain uniformity of method. Thus, if the general eq. be 



Ax 2 + By 2 + C + 2E# + 2Fy + G = 0, 

 then V=0 is the test of degeneracy. 



