122 Prof. Challis on the Zodiacal Light. 



light. I have given a proof of it in art. 15 of "The Theory of 

 Magnetic Force" contained in the Philosophical Magazine for 

 February 1861 ; which, as it lias probably attracted no attention, 

 I will here repeat, with some difference in the course of the 

 argument, and in somewhat more detail. 



The known general hydrodynamical equation applicable to 

 steady motion when no extraneous force acts, is 



« 2 .N ap .iog P =c-y, a) 



in which C is an absolute constant when the fluid is of unlimited 

 extent, as will be the case in the application we are about to 

 make of the equation. Consequently we obtain by differentia- 

 tion 



dp^_ __J_dY m 



pdx cl 1 dx 



Hence — /-, and, by parity of reasoning, — j- and — *£- are each 



pdx ' J r J pdy pdz 



of the order of the third power with respect to the velocity. For 

 cases of steady motion the general hydrodynamical equation which 

 expresses constancy of mass, viz. 



dp d . pu d .pv d. pw _ n 

 dt dx dy dz ~ ' 



becomes, to the second order of approximation, 



dx dy dz ' * ' 



because for that kind of motion -^-=0, and, as just proved, 



-j~, &c. are each of the third order. Now if u v v v w 1 ; u 2 , v^ 

 ax 



Wz, &c. be velocities due to different disturbances acting sepa- 

 rately and causing steady motion, each set of values will sepa- 

 rately satisfy the equation (2) . And from the form of the equa- 

 tion it is evident that if w=w 1 + w 2 H-&c, v = v } +v g + &c, 

 w = w x + w<2 + &cc., u, v, w will satisfy the same equation. That 

 is, the resultant of the separate steady motions, supposing them to 

 coexist, satisfies the equation. But since the component motions, 

 being steady, are functions of coordinates only, the resultant 

 motion must also be a function of coordinates only, and conse- 

 quently be steady motion. Hence the resultant will satisfy the 

 general equation (1) applicable to steady motion; and as it has 

 thus been shown that it satisfies both equations, the law of the 

 coexistence of steady motions is a necessary consequence. 



The application of this hydrodynamical law to the problem of 



