90 Prof. Challis's Researches in the 



whence, by substituting for -- , — , -.— , from the tliree first 

 equations, we liave the equation sought, viz. 



df^ ~^ ' \dx^ "^ df "^ dzV' 



(2.) The hypothesis of spherical waves. 

 In consequence of this hypothesis 5 is a function of the di- 

 stance (r) from the origin of co-ordinates, and the above equa- 

 tion becomes 



dKsr _ 2 d'^.sf 

 ~7F~""~d^' 



(3.) Integral of the last obtained equation. 



The general integral contains two arbitrary functions, each 

 of which separately satisfies the equation. I need only write 

 down the solution which has been employed in the present 

 discussion, viz. 



5=i.F(r-a/). 



(4.) Interpretation of the above integral. 



The meaning of this integral may be exhibited as follows. 

 Draw any straight line OABA'B' from O the origin of co- 

 ordinates, and putting z for r—at, describe a curve APB such 



that any ordinate MP shall represent the value of — -^ that 



is, of the condensation 5, corresponding to the value OM 

 of r. Because the function F is arbitrary, the form of the 



curve is arbitrary. It is admitted that the function F maybe 

 discontinuous, and accordingly that at a given time /„ Y{r — at^ 

 has real values from r=OA to r = OB, and that for values of 

 r less than OA and greater than OB, Y{r — at^ = 0. The 

 state of condensation which this curve represents is propa- 

 gated with the uniform velocity r/, so that at a subsequent 

 epoch ^2 it has the position of the curve A'P'B'. The change 

 which the curve has undergone is such that, A'B' remaining 

 equal to AB, any ordinate PM corresponding to an abscissa 

 AM has to the ordinate P'M' corresponding to an equal 

 abscissa A'M', the ratio of OM' to OM. Or if PM, P'M' 



