705] 
PROBLEMS AND SOLUTIONS. 
575 
system of all the substitutions (or say the entire group), or else it is a system the 
number of whose terms is a submultiple of the whole number of substitutions. The 
interesting question is, to determine those two or more substitutions, which, by their 
combination as above, do not give the entire group; for in this way we should arrive 
at all the forms of a submultiple group. 
[Yol. xv., p. 80.] 
3356. (Proposed by Professor Cayley.)—If the roots (a, /3, y, 8) of the equation 
(a, b, c, d, e) (a, l) 4 = 0 are no two of them equal; and if there exist unequal 
magnitudes 6 and </>, such that 
(6 + a) 4 : (6> + /3) 4 : (0 + 7 ) 4 : (0 + 8) 4 = (</> + a) 4 : (<f> + BY : (<£ + 7 ) 4 : (</> + S) 4 ; 
show that the cubinvariant 
ace — ad 2 — b 2 e — c 3 + 2bcd = 0; 
and find the values of 0, <f). 
[Yol. xvi., June to December, 1871, p. 65.] 
3507. (Proposed by Professor Cayley.)—Show that, for the quadric cones which 
pass through six given points, the locus of the vertices is a quartic surface having 
upon it twenty-five right lines; and, thence or otherwise, that for the quadric cones 
passing through seven given points the locus of the vertices is a sextic curve. 
[Vol. xvi., p. 90.] 
3536. (Proposed by Professor Cayley.)—A particle describes an ellipse under the 
simultaneous action of given central forces, each varying as (distance) -2 , at the two 
foci respectively: find the differential relation between the time and the excentric 
anomaly. 
[Vol. xviL, January to June, 1872, p. 35.] 
3591. (Proposed by Professor Cayley.)—If in a plane A, B, C, D are fixed points 
and P a variable point, find the linear relation 
a . PAB + B.PBC+ry.PGD + 8 . PDA = 0, 
which connects the areas of the triangles PAB, &c. 
[Vol. xviL, p. 49.] 
2652. (Proposed by Professor Cayley.)—Find the differential equation of the 
parallel surfaces of an ellipsoid.