Mr. G. G. Stokes on the 'Rectilinear Motion of Fluids. 297 



in which each of the six coefficients, the first, for example, ex- 

 presses a series of the form 



K*$+% +■$ + *« 



where X denotes the wave-length, and all the other quantities 

 are constant. The ellipsoid itself is the reciprocal of that 

 ellipsoid by which the wave-surface is constructed, and its 

 semiaxes are the three principal indices of refraction. As X 

 is supposed to vary, not only the length but the direction of 

 the principal axes vary, and thus we have a different wave- 

 surface for every different wave-length within the crystal. 



The optic axes are perpendicular to the circular sections 

 of the above ellipsoid, and describe, in general, two fragments 

 of a cone, the equation of which may be found by supposing 

 A to be variable in the equation of the ellipsoid. But only 

 very particular cases have been hitherto observed, and I shall 

 not stop to discuss them. 



Trinity College, Dublin, J. MacCullagh. 



September 1842. 



LI. Remarks on a paper by Professor Challis, " On the 

 analytical Condition of the Rectilinear Motion of Fluids." 

 By G. G. Stokes, B.A., Fellow of Pembroke College, Cam- 

 bridge*. 



TN the August Number of this Magazine (p. 101), Professor 

 *■ Challis has written an article, of which the object is to 

 prove that, in all cases of fluid motion in which udx + vdy 

 + wdz is an exact differential, the motion is rectilinear. The 

 importance of this question may apologize for these remarks, 

 since, if the reasoning in that article be correct, it will affect 

 the validity of much that has been written on the subject. It 

 appears to me however that Professor Challis has made an 

 assumption which is not allowable, and consequently the con- 

 clusion founded on it is not allowable either. In what fol- 

 lows, I shall call the path of a particle of fluid in space a line 

 of motion, and a line traced at a given instant from point to 

 point in the direction of the motion a line of direction. 



As the basis of his reasoning Professor Challis assumes, 

 that in every case where the continuity of the fluid is main- 

 tained, the most general supposition that can be made re- 

 specting the directions of motion in each indefinitely small 

 element of the fluid is, that they are normals to a surface of 

 continuous curvature, and as such pass ultimately through 



* Communicated by the Author. 



