34 



PROFESSOR CHALLIS, ON A NEW FUNDAMENTAL EQUATION 



7). 



6. It is next required to introduce into the equation of continuity of the fluid, by means 

 of equation (3), the condition of continuity of the motion. For this purpose the process must 

 be gone through which is given in my former paper (Camb. Phil. Trans. Vol. VII. Part III. 

 pp. 385, 386). The result there arrived at is, 



du dv dw _u dV 'j^ ^ w dV^ n 



Tw'^'d'y^l^^V''d^^V'dy'"V'dz'' \r '^ r . 



where r, r are the principal radii of curvature of the surface of displacement at the point xyz. 



u dx V dy w ds 

 V^Ts' V^Ts' V^Ts' 



dV be the increment of velocity along the line of motion corresponding to the increment ds, the 



required equation becomes for incompressible fluids, 



— + F - + - =0 (8). 



ds \r r ) 



When the fluid is compressible we have the equation, 



If ds be the increment of the line of motion, we have 



= -—. Hence if 



dp d.pu d.pv d.ptv 

 dt dx dy dz 



dp dp dp dp 

 or, -^ + ■^u + -fv + -£-w + 

 dw dy d« 



Now u = V 



,dx 

 dl' 



u= F 



dt 



dy 



fdu dv dw\ 

 "\dx dy d% I 



Az 



ds 



w = V^^ ; and, as before, 

 ds 



du dv dw dV ^IX l\ 



dw dy dm ds \r r ] 



By substituting these values in the equation above, it will readily be found that 



dp d.Vp 



dt 



ds 



"^e-?) 



0, 



.(9). 



in which d. Vp is the increment of Fp along the line of motion corresponding to the increment 

 ds of the line of motion. I have obtained equation (9) in my former paper (pp. .387 and 388) 

 by elementary considerations, and equation (8) might clearly be obtained in a similar manner. 

 That method, being independent, may be adduced in confirmation of the reasoning here employed, 

 and of the general equation (2), by means of which the reasoning has been conducted. It also 

 has the advantage of shewing distinctly that the increment d . Vp in (9) must be limited to 

 the direction of the line of motion, unless Vp has the same value at all points of a given 

 surface of displacement; and that rfF in (8) must be similarly limited, unless the velocity be 

 the same at all points of a given surface of displacement. 



The equations (8) and (9) may be called equations of absolute continuity. When they 

 are satisfied consistently with the respective dynamical equations, there can be no breach of 

 continuity and the motion is possible. Examples will hereafter be adduced to illustrate the 

 use of these equations. 



7. I propose now to determine by means of equations (s) and (9) in what cases of possible 

 motion udx + vdy + wdz is an exact differential. This important question has not yet received 

 a satisfactory answer.* 



" Lagrange in the Mecanique Analyliqne argues that udj;-\- 

 vdy + wdz is an exact differential when the motion is so small 

 that powers of the velocity above the first may be neglected ; 



and again, when the motion begins from rest. These theo- 

 rems occur in the Edition of Poisson's Traiic de Mecanique 

 of 1811, but are omitted in that of 1833. Lagrange's reasoning 



