and 



Prof. Challis on Hydro dijnamcis. 261 



do- da dr da rdO 

 ds ~~ dr ds rdd ds 



_da U da W 

 ~ dr'V + rdd' V 



dV_dU U rfW W 

 dt ~~ dt'Y + dt ' V' 



Hence, by substitution in the equation (7), 



The equation of constancy of mass to the same approximation is 

 da dV 2U dW W LQ n 



Now let it be required to ascertain from these two equations 

 values of U and W that are not related in any manner indicated 

 by the given conditions of the problem; but in a manner which 

 depends only on the mutual action of the parts of the fluid. 

 Such values must be found by integration, and it is therefore 

 necessary to obtain a differential equation proper for this pur- 

 pose, which may be done as follows. Since by hypothesis U 

 and W have no assignable relation, the factors by which they 

 are multiplied in equation (S) must be separately equated to zero, 

 and then from the two resulting equations and the equation (e) 

 U and W are to be eliminated. The equation which this pro- 

 cess gives is 



1 d^.ar d^.ar 1 /d 2 .ar d.ar , A rv% 



But for the solution of our problem this equation does not 

 possess the requisite generality. For not only may the values of 

 U and W depend on the mutual action of the parts of the fluid, 

 but also changes of these values from point to point accompany- 

 ing changes of condensation may be determined solely by the 

 same action. On this account I differentiate the above equation 



with respect to 6 in order to obtain an equation in which —^ 



is the principal variable. (It should be noticed that differentia- 

 tion with respect to r does not give in a similar manner an equa- 

 tion in which -j- is the principal variable.) After substituting 



q for -jq, the result of the differentiation is the equation which 



