9 , 
PEOPESSOE DONKIN ON THE ATTEACTION OE SOLIDS BOLESTDED 
constant independent of h, by taking a function of 4 as a parameter instead of A, we 
may suppose, without loss of generality, that this reduction has been effected. 
Mr. Cayley has shown*, in the latest publication on this subject which I have met 
with, that if two of the principal theorems of attraction (in the case of the ellipsoid) be 
given, the rest follow very simply, and are common to all suidaces of which those two 
can be predicated f. But the demonstration of the two assumed theorems constitutes 
the most essential part of the analytical problem, and it is my present object to show 
that they, and the others connected with them, are implied in the two differential equa- 
tions above written. 
1. Suppose the equation 
Y{x,y,zj)=0 (1.) 
represents closed surfaces for all values of the parameter 0 within certain limits ; and let 
the surface corresponding to any particular value of be called “ the surface 
With reference to these surfaces let the limits of integrals be indicated thus : [ signifies 
that an integral enclosed between the brackets, if it consist of superficial elements, is 
extended over the surface if of solid elements, through the whole space within that 
surface. In either case let [ ]*" stand as an abbreviation for [ ]^" — [ 
Let 
{(I)+(I)+(5)T 
be, for shortness, denoted by Q„ (whatever v may be). 
Let us put also 
and let the element of surface be called da. 
By means of (1.) we may suppose & expressed as a function of z. On this sup- 
position we have 
Let the sign of the right-hand member of this equation be so taken that Qfid shall be 
positive or negative according as the surface {d-\-d&) is without or within the surface 6. 
Since Qp'dd is the normal thickness at any point of the infinitesimal shell included 
between the two surfaces 6 and 0-\-d6, the latter surface must be either wholly within 
or wholly without the former, unless become infinite at some point of the sui’face 
a supposition which it is not necessary to exclude, but which I suppose to be excluded 
for the sake of simplicity. 
2. The preliminary propositions to be demonstrated in this article are not new. I 
am not certain to whom they are originally due ; they were, however, employed by Pro- 
* See also Camb. Math. Journal, vol. ill. p. 75 for the year 1842 . — Note added June 25, 1860. 
t Note on the Theory of Attraction (Quarterly Journal of Mathematics, vol. ii. p. 338). The two theorems 
are those marked V. and VI. in this paper. 
