MB. W. SPOTTISWOODE ON THE CONTACT OF CONICS WITH SURFACES. 303 
0 and T being two functions, with the exact forms of which we are not concerned. 
The expression then takes the form of the product of two factors, viz. (D'P — k&)t, and 
t 
tX-x T 
tY-yT 
tZ — zT 
u x 
P 
. 
• 
w' 
. 
P 
. 
v' 
, 
P 
= P 2 { P £ — t (u x X + w'Y -j- v'Z + l ,r Y) + T{u x x-\-w'y + v'z + I't ) } 
=P 2 £(P-P) 
= 0 . 
Hence u is a factor of (36). Now that equation as it stands is of the third degree in u ; 
so that u being divided out it is reduced to the second degree, say, to the form 
= Now referring to (31), and forming the equations in h, b, f, m ; g, . . ; 
1, . . , the equations corresponding to (36) may by a similar process be shown to be divi- 
sible by v, w, k ; and those divisions having been effected, the equations in question will 
be reduced to the forms \v 2 -\- ^+£, = 0 , . ., in which it is to be observed, from the 
symmetry of the expressions, that the coefficients of u 2 , v 2 , . . are the same. But as 
these equations (viz. those in u, v, w, k ) all lead to the same result, namely the deter- 
mination of the sextactic points, they can differ from one another only by factors. 
Hence, as in my memoir before quoted, equation ( 50 ), we must have identically 
, _ H-u + g _ ft|» + gi 
which can hold good in general, only in virtue of y being divisible by u, and g by u 2 , 
joq being divisible by v x and g, by v 2 , and so on. Hence (36) is divisible not only by u 
but by u 3 ; and that division having been effected, the degree of (36) will be reduced to 
12w-17. 
This completes the enumeration of the extraneous factors ; but although the degree 
of the equation (36) cannot in general be depressed below 12 n — 17, we have yet to show 
that, as stated in the introduction, the variables enter to the degree 10 in the form of 
lineo-linear functions of the arbitrary quantities. From what has gone before, it is clear 
that the quantities X, Y, Z, P, H involved in (36), are all functions of x, y, z, t, A, B, 
C, D only, that is to say, not of a, /3, . . a', /3', . . , except in so far as they are included in 
A, B, C, D. It remains to be proved that this is still the case after the differentiations 
and operation A involved in (36) have been performed. For this purpose let <p represent 
any function of x, y, z , t, A, B, C, D, and let . . indicate differentiation with respect 
to x, . . so far as they appear explicitly in p, irrespectively of A, B, C, D, then 
.'. wd I <p=A(ad A +/3d B +. .)p— a( Ad A +Bd B +. .)p + srd'<p ; 
