Geological Society. 1538 
functions of the second and third degrees respectively, and as regards 
the second system, for the fundamental symmetric functions or sym- 
metric functions of the first degree, and for the symmetric functions 
of the second degree in respect to each set of roots. 
“Memoir on the Resultant of a System of two Equations.” By 
Arthur Cayley, Esq., F.R.S. 
The resultant of two equations such as 
(a,'6;: ies y)"=0 
(n, % An y= 
is, it is well known, a function homogeneous in regard to the co- 
efficients of each equation separately, viz. of the degree n in regard 
to the coefficients (a, 6, . .) of the first equation, and of the degree 
m in regard to the coefficients (p, g, ..) of the second equation ; and 
it is natural to develope the resultant in the form AAP +4'A'P! + &e., 
where A, A!, &c. are the combinations (powers and products) of the 
degree n in the coefficients (a, b,..), P, P’, &c. are the combinations 
of the degree m in the coeflicients (p, q, . .), and /, k', &e. are mere 
numerical coefficients. The object of the present memoir is to show 
how this may be conveniently effected, either by the method of sym- 
metric functions, or from the known expression of the resultant in 
the form of a determinant, and to exhibit the developed expressions 
for the resultant of two equations, the degrees of which do not ex- 
ceed 4. With respect to the first method, the formula in its best form, 
or nearly so, is given in the ‘ Algebra’ of Meyer Hirsch, and the appli- 
cation of it is very easy when the necessary tables are calculated : as 
to this, see my ‘‘ Memoir on the Symmetric Functions of the Roots 
of an Equation.” But when the expression for the resultant of two 
equations is to be calculated without the assistance of such tables, it 
is, I think, by far the most simple process to develope the determi- 
nant according to the second of the two methods. 
GEOLOGICAL SOCIETY. 
[Continued from p. 74.] 
June 17, 1857.—Col. Portlock, R.E., President, in the Chair. 
The following communications were read :— 
1. ‘‘On some Comparative Sections in the Oolite and Ironstone 
Series of Yorkshire.” By John Phillips, M.A., F.R.S., F.GS., 
Reader in Geology in the University of Oxford. 
The author first called attention to the problem presented to geo- 
logists for solution, which is suggested by the extremely unlike 
mineral aspect, and only partial agreement in fossil contents, of the 
two separated oolitic tracts of North Yorkshire and the south of 
England. He showed on what principles W. Smith proceeded in 
1817 and subsequent years to investigate the contemporaneous lines 
in these two tracts, and explained his own researches in the same 
direction in 1824 and following years. ‘lhough in regard to the 
main determinations arrived at and published by himself in 1829 
