394 PROFESSOR CHALLIS, ON THE DIFFERENTIAL EQUATIONS 



ADDITIONAL NOTE. 



The following is a proof of the Proposition enunciated in Art. 6, viz. 

 that when udx + vtly + wdz is an exact differential the velocity does 

 not vary from one point to another of a surface of displacement. 



When the continuity of the fluid is maintained, the most general 

 supposition that can be made respect- 

 ing the directions of motion in each 

 indefinitely small fluid element is, that 

 they are normals to a surface of con- 

 tinuous curvature, and consequently 

 that they pass ultimately through two 

 focal lines perpendicular to their di- 

 rections and situated in planes at right angles to each other. 



Let, therefore, PWO, pqr be straight lines drawn in the directions 

 of the motion at a given instant at two points P, p, of an indefinitely 

 small element, and let them pass through the focal lines Wq, Or. The 

 point P is referred to the rectangular axes AX, AY, AZ. AM = X, 

 MN = Y, NP = Z. Pp is drawn parallel to AX and is equal to §X. 

 Let OW — I, WP = r, and draw ps perpendicularly on OWP. Then 

 Ps = Sr. Take now another system of rectangular axes Ox, Oy, Oz, 

 of which Ox coincides with OWP, Oz with the focal line Or, and 

 Oy is parallel to Wq. Let Or = h, Wq = k. Suppose the equations 

 of Pp referred to the axes Ox, Oy, Oz, to be x = az + a, y = bz + (S. 

 Because it passes through P, the co-ordinates of which are x = I + r, 

 y = o, z = 0, it follows that a = I + r, and (Z = 0. Hence the equations 

 become x = az + I + r and y = bz. 



Again, let the equations of pqr be x = mz +?i, y =pz + q. As this line 

 passes through the point r, the co-ordinates of which are x = 0, y = 0, 

 z = h, we have = mh + n and 0=ph + q. Therefore x = m(z — h), and 

 y = p (z — h). Since also the line passes through q, the co-ordinates of 



