Prof. Cayley on Differential Equations and Umbilici. 451 



the two tangent planes through (x=0, y = 0) are given by the 

 equation x 2 + y 2 =0; and for these planes we have p 9 -\- 1=0. 

 The factor p 2 + 1 = determines therefore the directions of the 

 envelope at the conical point. 



VIII. 



In verification of the equation 



z = ik(x^ + y^)+^ax(x 2 + y 2 ) 



for a quadric surface in the neighbourhood of the umbilicus, 

 I remark that, starting from the equation 



a 2 + b 2 I" C 2 



of an ellipsoid, and taking u, 0, 7 as the coordinates of the um- 

 bilicus, and as the inclination to the axis of x of the tangent 

 to the principal section through the umbilicus, then transforming 

 to the umbilicus as origin and the new axes through that point, 

 viz. the axes of x, z being the tangent and normal in the plane 

 of ac, and the axis of y being at right angles to this or in the 

 direction of b, the equation becomes 



(a + #cos# — zsinO) 2 y 2 (y—xsm0—zcos0) 2 _ 



? + P + ? ~~ "- 1 ' 



or, expanding, 



— 2zxsm 0cos0(— 2 2J=0» 



But we have 



and thence 



tan0=£v^± 2 , 

 a^/^-c 2 



A cs/a 2 -b 2 c ' a^/b 2 -c 2 a 



SID U= , , z=.—u COSfl= , . = — ry; 



b^ a 2 -c 2 oa b^/ a 2 -c 2 be y ' 



and substituting these values, the equation becomes 



~ 2z 7a + ¥ + ¥ +Z <MV +3 f^~ -ZX=:0, 



