248 
ME. W. SPOTTISWOODE ON MULTIPLE CONTACT OE SUEFACES. 
but not the relations 
^=10 : 1“ -1 0=12 : 1” _1 2= ... 
Consequently we cannot eliminate, in the manner there effected, the quantities relating 
to the quadric V from any of the six equations derived from the forms U = 0, □ U = 0, 
□ 2 U=0. We may, in fact, by means of these equations, determine six out of the nine 
constants of V, but that is all. 
When, however, P is a quartitactic point, we have the four additional equations, say 
□ 2 3 3 u=o, □» 1 u=o, □? a u=o, □; 4 u=o, 
ten in all. From one, or two, or three of these we may, by means of the equations 
□ 2 2 3 u=o, □ 2 1 u=o, □; a u=o, 
eliminate one, or two, or three of the quantities 6 : 23, & : 31, 9 : 12, and obtain conditions 
of the form 
3A 2 U . BAU— A 3 U . BU=0, 
to which the proper suffixes 23, 31, 12 are to be appended. 
By this means we may either determine all nine of the coefficients of the quadric, and 
have one condition of the above form ; or we may determine eight of the coefficients, and 
have two such conditions ; or, lastly, we may determine seven of the coefficients, and 
have three such conditions. 
From this it would appear that, if we regard the conditions as equations in the coor- 
dinates of the point P, there will be a curve on U every point of which will be quar- 
tactic, with a single quadric at each point. Again, there will be a definite number of 
points which will be also quartactic, having a singly infinite number of quadrics having 
four-branch contacts at the points. 
When P is a quintactic point, we have the five additional equations, 
□ 2 4 3 u=o, n^u=o, □t 2 u=o, n* 4 u=o, n 2 4 4 u=o. 
From three, or four, of these we may eliminate three, or four, of the quantities 
9 : 23, 9 : 31, 9 : 12, 9 : 14, by means of the equations 
□lsU=o, niJj=z o, d 2 2 u=o, □? 4 u=o, 
and obtain results of the forms 
(BU) 2 A 4 U — 6BU . A 2 U . BAU+3(A 2 U) 2 (A 2 +B 2 )U=0, 
3(BAXJ) 2 A 4 U — 6BAU . A 3 U . BA 2 U-f(A 3 U) 2 (A 2 +B 2 )U=0. 
If we have previously determined the nine coefficients, or if we have determined 
eight only, and use one of the new equations for determining the ninth, or if we have 
determined only seven, and use two of the new equations for determining the two 
remaining coefficients, we shall have six conditions ; or, lastly, if having determined 
seven only, we now determine one more, we shall have seven conditions. 
