OF HEAT IK ELLIPSOIDS OF REVOLUTION. 
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ellipsoid whose major axis is that of revolution, it is proved that the expressions and 
results may be easily transferred to the case of a planetary ellipsoid. And as the chief 
object of the paper is to examine the forms and properties of the functions which 
present themselves in the mathematical treatment of the subject, I have not thought 
it necessary to enter on the discussion of the movement of heat in a shell bounded by 
two confocal ellipsoids, or in an infinite solid from which an ellipsoidal cavity has been 
taken—questions, however, which might be easily treated by introducing the functions 
T„ as well as S„. The analysis would seem also to be applicable to other physical 
problems relating to ellipsoids of revolution. 
Perhaps I should explain that if I have entered somewhat minutely into points 
which are not new, I have done so for the purpose of rendering the argument clearer 
and more coherent. 
2. When a solid body is heated in any manner and left to cool, the equations which 
have to be satisfied are, first, the general equation of conduction 
d 3 V d 2 Y d 2 Y 1 dV 
c far - *" dy~ dz 2 t dt . 
wherein V is the temperature at any point, f the “thermal diffusivity” of the body; 
2° the boundary condition, which we shall suppose to be one or other of the forms 
V=0or^+W=0.(2) 
wherein l~ +ra^ + n~, i m n being the direction cosines of the normal N to 
cfN do: dy dz ° 
the surface measured outwards, and f) is a constant. 
The first of these conditions corresponds to the case where the surface of the body 
is maintained everywhere at uniform temperature, for we may suppose the zero of 
temperature so chosen as to coincide with the given temperature, whatever it may be. 
The second is the usual condition of radiation according to Newton’s law of cooling. 
We have also a third condition, that which gives the initial state of the body :— 
V = V 0 =y'(a;, y, z), when t — 0.(3) 
The general course of solution, whatever the sohd may be, is to put 
Y=tA.e~ tK ' / .v .(4) 
where v is a function of x y z, so chosen as to satisfy the equation 
