ME. W. SPOTTISWOODE OX THE CONTACT OE SUBFACES. 
263 
Similarly, if U 0 , U), U 2 be the same functions, the first quadratic in a, j3, y, c$; the 
second lineo-linear in a, (3, y, h ; a', (3 1 , y', b' ; the third quadratic in a 1 , (3 1 , y 1 , h', say, 
U 0 = (a, /3, y, b) 2 , 'i 
U ,=(a,{3,y,b)(a',(3', 7 ',h')S (15) 
TJ 2 =( a ',(3',y',V), J 
then it will be found that 
(U 0 , U„ U f ), («', — a) 2 =( . c, —b,f)\ \ 
(U 0 , U„ U 2 ), (13', —[3) 2 =( — c, . a, g)\ 
(U 0 , U„ U a )(/3', — fi)(y', —y)=(—c, . a, g)(b, —a, . h), 
(U 0 , U„ U 2 )(a', -a)(S', — &) =( • C, — &,/)(/, g, K .). 
. . . (16) 
And a similar process is also obviously applicable to functions of higher degrees. 
§ 2. Conditions of Contact. 
In the memoir “ On the Contact of Conics with Surfaces,” above quoted, it was shown 
that the conditions for a 1, 2, . . pointic contact at a point P of the curves of section of 
the surfaces U, V, made by a plane Aa’-j-Py-j- Cs + 1J£ = 0, may be expressed as follows: 
V=0, [=lV=0, n'A T =0.., (17) 
The cutting plane, say the plane (A, B, C, D), was supposed to pass through the 
point (ay g, z, t), say the point P, to be capable of revolving about the axis whose six 
coordinates are ( a , b, c, f g , h), and to have an azimuth which, measured about the 
axis, is determined by the quantity ■&' : 
The equations (17) may be supposed to be expressed in any of the forms to which 
they were reduced in § 1. Taking, for instance, the form (8), and dropping for the 
present the suffixes, so that □ , □ ' shall be understood to represent any pair of the 
operators □ the equations (17) may be written thus: 
v=o, (zn'u — wn')Y=o, (ot'd — z*n') 2 v=o ( 18 ) 
In the expansion of these expressions there will occur combinations such as oAV, 
where i and j represent any of the numbers 1, 2, 3, 4; and when i and j are different, 
the compound operation ^ is not in general the same as hf. But in the case where 
e>,V =0, eSyV=0, it will be found on actual trial of the special forms that ^ f V=^.V. 
The same thing may be also proved by the following considerations, which are appli- 
cable to all the forms of ^ and o ; . In the first place is symmetrical, except as 
regards algebraical sign, in respect of the first differential coefficients of U and V ; so 
that if % be the expression obtained by replacing the differential coefficients of U by 
