Prof. Challis on a Mathematical Theory of Heat. 207 



to the square of the velocity of the secondary waves, and there- 

 fore to the square of the velocity of the original waves, which 

 latter velocity varies inversely as the distance from a remote 

 origin, it follows that the heating effect of a radiating hot body 

 upon another distant body varies inversely as the square of the 

 distance between them, as is found by experiment to be the 

 case. 



If the investigation had taken account of terms of the second 

 order in the value of the velocity of the secondary waves propa- 

 gated from the atom A (as terms of that order were taken into 

 account in the value of the condensation), the effect on the 

 motion of translation of the atom A' would have been of the 

 order of the fourth power of the incident velocity, and may 

 therefore be neglected in comparison with the effect due to 

 terms of the first order. 



The part of the velocity accompanied by condensation, which 

 was left out of account in the preceding investigation, has at 



the surface of the atom A the ratio -~ to the part considered, 



A, 



/S being the radius of the atom. This ratio is so small that the 

 effect of the omitted part may be wholly inappreciable. It is, 

 however, to be observed that the velocity accompanied by con- 

 densation varies inversely as the distance, and consequently that 

 its moving effect on the atom A' would be found by a like in- 

 vestigation to vary inversely as the square of the distance. Also 

 the condensation accompanying it would cause a propagation of 

 velocity to the opposite side of A', so that the integration with 

 respect to Q would have to be taken to some value greater than 



-' and the effect of the velocity on the side towards A would be in 



part counteracted. It is not my intention to enter fully upon the 

 consideration of the waves of condensation in this communica- 

 tion ; but an important general remark relating to them may be 

 appropriately made here. 



If we suppose waves of condensation to be propagated from 

 all the atoms contained in a spherical space of radius r, the con- 

 densation resulting from the composition of the waves at any 

 distance (II) from the centre of the sphere very large compared 

 to r will vary as the number of atoms, that is, as r^ directly, 

 and as K inversely, the mean interval between the atoms being 

 given. If now we take another sphere, of larger radius, ?', and 

 another point, whose distance (iV) from the centre of the sphere 



is such that it = fi?* then the condensation at the first point is 



