Prof. Cayley on a Quartic Surface. 19 



a triangle similar and similarly situated to that described above 

 without the help of an equation. 



In the case of direct combination, the visible branch is beyond 

 the wall, and is convex to the observer; the other branch 

 passes through the eyes ; and a short arc, generally near the 

 apse, is in front of them, but, being formed by combining points 

 of which one is concealed from the corresponding eye, is invi- 

 sible. The rest is behind the observer, and I suppose none of it 

 can be seen in its proper place ; but a small part near the second 

 asymptote is formed by combining points, both of which may be 

 visible, and under favourable circumstances might be combined 

 by divergent visual rays. Where the image would appear, is a 

 curious question. 



In the case of converse combination, the visible part is on this 

 side of the wall, and is of course concave to the observer. The 

 curve runs up to the nearest eye, and, though it comes out at the 

 other, is seen no more of. ■...., 



Gibraltar, November 1864. 



IV. Note on a Quartic Surface, By A. Cayley, F.R.S* 



IT would, I think, be worth while to study in detail the quartic 

 surface which is the envelope of a sphere having its centre 

 on a given conic, and passing through a given point. The equa- 



tions of the conic being z = 0, -g +p=l, the coordinates of a 



point on the conic may be taken to be % = acos 6, y = bsm0j 

 z=0, whence, if (a, /3, 7) be the coordinates of the given point, 

 the equation of the sphere is 



(%-acos6f+(y-hsmd)* + z*={a-aco$6)*+{/3-bsm0)2 + y*, 

 or, what is the same thing, 



a .* + y« + ^-««~ / S«- 7 «-2(a?-a)flco8^-%-i9)&8in^ = 0; 

 and hence the equation of the surface is at once seen to be 

 (a.« + y* + ^-a«--)S9- y a)« = 4fl 2 («— a) 2 + 46 9 (y-/S) 8 . 



If a = b (that is, if the conic be a circle), then we may without 

 loss of generality write /3 = 0, and the equation then is 



(^ 2 H-?/ 2 + ^ 2 .-a 2 -7 2 ) 2 = 4« 2 |(^-a) 2 + ^ 2 |. 



This may be written 



/ 2 ■ 2 . 9 2 2 o 2\2 a 2-Y 2 -f 2a 2 + 7 2 — ^ 2 \ , 

 (# 2 + «/ 2 + £ — or — <y i —2a i y= —-Sara (a?— — — - — '— 1, 



* Communicated by the Anther. 

 C2 



