534 Prof. Challis on the Principles of Hydrodynamics. 



mals will meet two and two in two focal lines, situated in planes 

 of greatest and least curvature, and cutting the normals at right 

 angles. Let the small area of which P is the centre be m^, and 

 let r, r^ be the distances of the focal lines from P. Then if Br 

 be the given small interval between the surfaces, the area on the 

 other surface formed by joining the points of its intersection by 

 the normals, is ultimately 



{r + Sr){r' + 8r) 



rw 



But as the dii-ection of motion through P is in general continu- 

 ally changing, the position of the surface of displacement through 

 that point will vaiy with the time. Hence the positions of the 

 focal lines and the magnitudes of r and i^ will change continually, 

 whilst the area m^ may be supposed to remain the same. Let r 

 and r' represent the values of the principal radii of cm-vature at 

 the time t, and let « and ^ be the velocities of the focal lines 

 resolved in the direction of the radii of curvature, and considered 

 positive when the motion is towards P. Then at the time t-\-Bt 

 the values r and r' become r — aht and r' —^Bt, and the elemen- 

 tary area on the second sm'face is 



2 {r + Sr-oiBt){r' + Br-^St) 

 "" ■ {r-ccBt){r'-^8t) 



which, by omitting quantities of an order that may be neglected, 

 is equal to 



„«. fc±M^. (] + ?^'-) (i + §^). 



Hence by rejectitig small quantities of the second order, a and 

 /3 disappear, and the result is the same as if the position of the 

 focal lines had been supposed fixed. If therefore V and p be 

 the velocity and density of the fluid which passes the area m^, 

 and V and p' be the velocity and density of the fluid which 

 simultaneously passes the other area, and if these qualitities be 

 supposed constant during the small interval Bt, which is allowable, 

 the inctement of matter between the two areas in that interval is 



2 (r + Br){r' + Br) ,^j,^ ^ „jj 

 —m^. -^1 -p\'ot-\-m^.pybt. 



And this quantity must be equal to m^-j^BtBr. Hence 



m 

 or 





i 



