118 
PROFESSOR C. NIVEN ON THE CONDUCTION 
and for the other odd. The values of /2(a) divide themselves in like manner into two 
classes. These values are expanded, in the first instance, in terms of the minor axis 
of the confocal ellipsoid, and afterwards in terms of the major axis, the former series 
proceeding by functions which satisfy the differential equation 
cl 2 u 2 du ( n.n + 1 
rh ~ ' .?■- dr, \ 1 x~ 
u = 0. 
Of the two solutions of this equation, which are both finite in form, one S„ is finite 
when x = 0, while the other T« becomes infinite. 
The expression plays for spheres the same part that Bessel’s function plays for 
circular cylinders, and as Heine has employed the term cylinder-function for the 
latter, it would seem consistent with analogy to use the term spherical-function for 
S,„ When n is expressed in terms of the major axis of the confocal ellipsoid two 
expansions are given, one in terms of spherical functions and the other in terms of 
associated functions, and it is shown that each of these series possesses special 
advantages in relation to particular points which arise in the problem of the con¬ 
duction of heat. 
The properties of these functions are afterwards further considered. 
The above expansions being found, I have next discussed the system of equations 
which determine k, following more or less closely the analysis which Heine has given 
of a similar system, and it is proved that the values of k are all real and definite in 
position, and that for these values the expansions of 3 and fl converge rapidly when 
a sufficiently large number of terms is taken. 
I then show how to express, by successive approximation, the roots of the equation 
in k in powers of a quantity e which depends on the eccentricity, and have entered 
with some fulness of detail into the numerical calculation of a few of the smaller roots 
and of the corresponding coefficients of the functions 3- and /2, more especially in the 
case of the first of the two classes into which they fall. Besides these particular 
values, however, the general formulae are given, which will furnish them to a certain 
degree of approximation for all values of m and for any value of k. 
With regard to the special problem of the conduction of heat, the boundary con¬ 
dition is supposed to be either that the surface is kept at a constant temperature, or 
that the body is cooling by radiation. The former is mathematically the simpler, and 
we might imagine it realised in the case of a body kept in the midst of an infinite 
fluid after a sufficient time has elapsed for the surface to take the temperature of the 
fluid. With this assumption the different values of X might be found from the equa¬ 
tion /2,/—0; and the roots of /2 0 °=0 are investigated up to e 4 . The general condition 
of radiation is next considered, and it is shown how it may be brought theoretically 
within the range of analysis. I have not, however, thought it necessary to do more 
in this direction than indicate how the successive approximations may be found. 
Although the calculations were undertaken in the first instance for the case of an 
