332 Mr A. R. Forsyth, On Systems of Quaternariants. [Feb. 11, 
properly chosen and absolutely independent of one another. The 
leading coefficient of a quaternariant satisfies three linear partial 
differential equations, also of the first order; and when obtained, 
it uniquely determines the quaternariant by being symbolized 
into the umbral elements of the coefficients of the quantic,—this 
result being a consequence of the theorem proved that every 
quaternariant is expressible as an aggregate of symbolic products 
of factors of some of five forms. 
The number of quaternariants in an algebraically complete 
system is V—5, where WV is the number of coefficients in the 
most general form of the quantic (or of the set of simultaneous 
quantics) with which the system is associated. A method is 
indicated by which the leading coefficients of these V —5 quater- 
nariants can be obtained as combinations of binariants, which 
belong to bimary quantics derivable from the original quantic. 
And for the quaternariants which do not involve line-variables 
and which belong to unipartite quantics in point-variables, it is 
shown that their leading coefficients can be expressed as contra- 
variants (and invariants) of ternary quantics derivable from the 
original quantic. 
The general theory thus indicated is applied to obtain the 
special results for the following cases: (i) a quadratic: (411) two 
quadratics in point-variables: (111) a lineo-linear quantic in point- 
and plane-variables: (iv) a linear complex: (v) a congruence of 
two linear complexes: (vi) a regulus of three linear complexes: 
and (vil) a quadratic complex. 
(2) On the stresses in rotating Spherical Shells. By C. 
CuREE, M.A., King’s College. 
[ Abstract. ] 
IN a previous paper the author obtained data from which a 
complete solution of the problem of a rotating isotropic spherical 
shell was deducible, but considered in detail only the cases of 
a solid sphere and of an extremely thin shell. In the present 
paper the general solution, applicable whatever the thickness of 
the shell, is given explicitly. ’ 
This solution is so complicated that in its applications it is 
essential for clearness to select representative materials, in which 
there is a necessary relation between the two elastic constants 
of the orthodox British, or bi-constant theory. ‘The relation 
mainly considered is that which on the “ uniconstant” hypothesis 
is necessarily true, and is accepted as such by most foreign 
elasticians. The limiting case when Young’s modulus is thrice 
