426 The Rev. Prof. Challis oti the Necessity of Three 



focal lines will vary as well on this account as on account of 

 any changes in the magnitudes of r and r'. Let a and /3 be 

 the velocities of the focal lines estimated in the direction of the 

 radii of curvature, and considered positive when their motions 

 are towards P. Then, at the. time ^ + S /, the values of r and 

 r' become r—ctZt and r' — /3 It; and assuming the area m^ to 

 be constant, the elementary area on the second surface is 

 2 {r + lr-alt){iJ-\-lr-^lt) 

 ^ ' {r-udt){tJ-fnt) * 



Hence, by omitting quantities of the order x — , the re- 

 sult is the same as if the positions of the focal lines and mag- 

 nitudes of the radii of curvature had not varied. If now V and 

 Q be the velocity and density of the fluid which passes the area 

 m% and V and g' the velocity and density of the fluid which 

 simultaneoush' passes the other area, and these quantities be 

 supposed (as is allowable) to be uniform during the time 8 if, 

 then considering the velocity positive when directed from the 

 focal lines, the increment of matter between the areas in the 

 time 8 ^ is 



— m^ .- -^ -q'\'ot + m^q\oti 



or 



-'«'^'fil? + ^Hv + F)}^- 



by neglecting quantities of an order already neglected. If 8^ 

 be the increment of density in the time 1 1^ the increment of 

 matter is also m^tg^r. Equating these two values and passing 

 from differences to differentials, we have 



^-f + '^? + ^v(l + J,)=0. . . . (A.) 



We have thus been led by elementary considerations to a 

 general hydrodynamical equation. On reviewing the fore- 

 going reasoning it will be seen to rest on two principles, on 

 the constancy of mass of each elementary portion of the fluid, 

 and on the existence of surfaces normal to the directions of 

 motion and possessing the properties of surfaces of continuous 

 curvature. It is also assumed in the course of the reasoning 

 that two such surfaces indefinitely near each other are ulti- 

 mately parallel, and (since a and /3 are finite velocities) that 

 the radii of curvature of a surface passing through a given 

 point do not vary abruptly^ either in magnitude or direction. 



