592 
DE. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
16. The equation (26) of can be written in a more suitable form by means of the 
anharmonic angles of art. 9 ; for this purpose let us denote 
✓u-gn W - bycosL; b ? - cosL ’ &c - 
Let us denote also S' by cos 2 ^, S" by cos 2 */', &c. It is evident that the angles L, L', g', 
g", &c. may be either real or imaginary; when the substitutes are made, we get 
Jj=^ 2 sin 2 g' -\-y 2 sin 2 g" -\-z~ sin 2 g"'-\-w* sin 2 g'" 
+ 2 yz sin g" sin g'" cos L + 2 zx sin g'" sin g' cos M + 2 xy sin g' sin g" cos N 1. (27) 
+2 xw sin g' sin g"" cos P+2 yw sin g" sin g'" cos Q,-\-2zw sin g'" sin g"" cos R=0. J 
Compare equation (3), art. 3. 
Cor. If the four quadrics S^— A, S= — B, &c. be mutually orthogonal, the equation of 
their orthogonal quadric will he Jj=^ 2 sin+'+j 2 sin 2 g>"+z 2 sin 2 */"+w 2 sin+"", or of the 
form a ,2 +++^ 2 +w 2 = 0, and there will be only one orthogonal quadric instead of eight. 
Observation. This section is abridged from a Memoir by me in Tortolini’s ‘ Annali di 
Matematica,’ serie ii. tomo ii. fasc. 4. 
CHAPTER II. 
Section I . — Generation of Cy elides. 
17. We have seen that a cyclide is the envelope of a variable sphere whose centre 
moves along a given quadric, and which cuts a given sphere orthogonally (see art. 5). 
I shall call the variable sphere the generating sphere (a name which I find more con- 
venient than enveloped sphere or enveloppee), and the quadric which is the locus of the 
centres of the generating sphere I shall call the focal quadric , a term expressive of an 
important property which it possesses. In De la Gournerie’s Memoir “ Sur les lignes 
Spheriques,” he uses the name defer ente in an analogous case (see Liouville’s Journal, 
1869). The sphere which is cut orthogonally we shall call the sphere of inversion, and 
it will be always denoted by the letter U, unless the contrary be stated, and the focal 
quadric by the letter F. 
18. If we draw any tangent plane to F this will intersect F in two lines, the gene- 
rating lines of F at the point of contact. Now let us conceive three spheres whose 
centres are at the intersections of these lines and at a consecutive point on each line 
respectively ; then if they cut U orthogonally, the two points common to the three will 
evidently be their points of contact with their envelope. Now it is evident that these 
points are the limiting points of the system composed of U, and the tangent plane to F. 
Hence we have the following method of generating cyclides : — 
Being given a quadric F and a sphere U, draw any tangent plane P to F, the locus of 
the limiting points of U and P will be a cyclide. 
