490 Prof. Challis on tlie Dispersion of Light. 



is incident in the direction of the axis of a? on a fixed smooth 

 sphere of given radius ; it is required to find the condensation 

 at any point of the surface of the sphere at any instant. Accord- 

 ingly the Article in the June Number consists almost exclusively 

 of investigations preparatory to the solution of this problem. In 

 the first place the solution was attempted by employing only the 

 two usual fundamental equations, and results were arrived at 

 which were subsequently shown to be incompatible with the given 

 conditions of the problem. The third general equation was 

 then taken into account, and another process of solution, in- 

 volving the principle of the new equation, was then entered 

 upon, but not brought to a conclusion. I propose now to 

 resume the argument, taking it up from the point (in p. 458) 

 at which the second process was commenced. 



It is unnecessary to repeat that part of the reasoning (ending 

 in p. 461) by which the equation 



o a da #v n , v 



Ka -lk + W= ' w 



was obtained. It will suffice to explain here that in this equation 



a is the condensation, and V the total velocity, at any point 



whose coordinates are %> y, z at the time t ; da is the increment 



of condensation, at the given time, along the line of motion 



passing through that point, corresponding to the increment ds 



dV 

 of space along the same line ; and -r- is the partial differential 



coefficient of V with respect to t. The factor /e 2 # 2 , which holds 

 the place of a? in the usual mode of investigation, takes account 

 of the composite character of the motion. The equation only 

 embraces quantities of the first order, and is exclusive of the 

 action of any extraneous force ; in other respects it is perfectly 

 general. For our present purpose we have to apply it to cases 

 in which the motion is symmetrical with respect to an axis. 



In such cases, if r be the distance of any point from the 

 origin of coordinates, and 6 the angle which this line makes 

 with the axis of oc, a and V are functions of r and 6. Also, if 

 U and W be the resolved parts of the velocity respectively along 

 and perpendicular to the same line, V 2 = U 2 + W 2 . Consequently 

 we have the equations 



da _JLa dr .da rdd 

 ds ~~ dr ds rd0 ds 9 

 _da U da W 

 ~dr'V + rd0'Y' 

 and dV_dV V.dW W 



dt~~ dt ' V + dt ' V ' 



