590 THE HON. J. W. STRUTT ON THE VALUES OF A DEFINITE INTEGRAL. 
The same thing is true for an ellipsoid or body of any figure which lies altogether on 
one side of every tangent plane, namely, that the terms of odd order in H (except one) 
are wholly without influence on it, and for the same reason. 
We saw that in the case of a sphere the mean temperature was independent of the 
conductivity, and also of the size of the sphere; but this depends on Newton’s law of 
cooling. A comparison, however, may be made which shall hold good whatever be 
the law of variation of radiation with temperature ; for if the conducting-power of any 
uniform body (which need not be oval) be increased in the same proportion as its linear 
dimensions, a corresponding distribution of temperature will satisfy all the conditions. 
Conclusions of interest from a physical point of view may be deduced from the foregoing 
considerations, but I refrain from pursuing the subject at present, as the physical problem 
was only brought forward in illustration of the mathematical results developed in this 
paper. 
of even order has identical values at opposite points of the sphere, and one of odd order has contrary values. 
The product of two harmonics which are either both even or both odd has therefore the same value when inte- 
grated over any portion of the sphere, or over what may be called the opposite portion, or as a particular case 
over two opposite hemispheres. The last two integrals are the halves of the integral over the whole sphere, 
which vanishes by a well-known property of these functions. 
