388 



Mr. W. H. L. Russell. 



[June 21, 



XIII. " Theorems in Analytical Geometry." By W. H. L. 

 Russell, F.R.S. Received June 21, 1888. 



To determine the envelope of the first polar of any curve, when the 

 pole moves on a given curve of the third order. 



d~F 



Let F (£, 7j, £) = be equation to the surface ; then if p = — i 

 d F c?F 



g[ = -^r-, r = — , j>» + gfy + rz = is the equation to first polar, 

 when (x, y, z) moves on a given cubic, 



x 3 + y s + £ 8 + Gmxyz = 0. 

 Then differentiating 



(a^-f 2myz) dx + (?/ 2 + 2?7ia?«) d«/ + (z 2 2mxy) dz = 0. 

 pefa? -f qdy + rcZz = 0. 



Then as usual 



# 2 + 2myz = X-p, 



?/ 2 + 2mxz = \q, 



z 2 -f 277i«y = Xr. 

 Then eliminating z by the equation to the first polar, we have — 

 Ax 2 + Bxy + Cy 2 — p, 

 Dx* + Exy + F^/ 2 = q, 

 Gx\+ Rxy + Kt/ 2 = r. 



where A, B. ..." . are fun oions of pqr, whose forms are imme- 

 diately seen, and the arbitral t multiplier is omitted because ifc will 

 disappear in the final result: then we find at once the values of x 2 , xy 

 and so z 2 , y 2 , and therefore of x, y, z, which we may substitute in the 

 equation to the polar, and so obtain the envelope. But we may find a 

 more symmetrical result thus : eliminating as before by means of the 

 equation to the polar — 



My 2 + B'ijz + C'z 2 = p, 



D'y 2 + E'y* + F'z 2 = q, 



Gy + H'^ + K'^ = r; 



and moreover 



