Condition of the Rectilinear Motion of Fluids. 103 



O z be the axes of rectangular coordinates, and let the coor- 

 dinates O M, M Q, Q P of the point P be x, y, s t and the 



coordinates O m, m q, qp of p be x + dx, y, z; so that the 

 indefinitely small line Pp is parallel to the axis O x. Draw 

 the straight lines P N A, p n a, in the directions of the motion 

 at the points P, p, at a given instant. Since these points are 

 supposed to be indefinitely near each other, they may be con- 

 sidered to belong to the same indefinitely small element of the 

 fluid, and consequently, by what has just been said, the lines 

 PNA,|)nflj are ultimately normals to the same curve sur- 

 face, and pass through two focal lines such as N n and A a. 

 Take A, the intersection of P N A with the focal line A a, 

 for a new origin of rectangular coordinates x p y p z t ; and let 

 the axis A z, coincide in direction with A «, the axis A x, with 

 A N P, and the axis Ay t be parallel to N n. Draw p s per- 

 pendicularly on A x r Let A N = /, NP = r, and P s = r { . 

 Also let the velocity at P be V, and that at the same time at 

 p be V + V,. 



The component of the velocity at P in the direction of z 

 being w, let the component of the velocity at p in the same 

 direction be w + dw. Then, 



w = V cos <APQ, 

 and w -f d *w = (V + V,) cos <apq f 



= V cos <apq + V, cos < A P Q, 

 terms of the second order being neglected. Hence 



dio= V (cos <ap q — cos < A P Q) + V y cos < A P Q. 

 Also, d x = P s sec < p P s = r, sec <p P s 3 



therefore 

 dw __ V (cos «< a p g — cos < A P Q) + V/ cos <; A P Q 

 d x ~" r t sec < p P s 



