298 Mr. G. G. Stokes on the Rectilinear Motion 



two focal lines ; that is to say, that the above is true neglecting 

 quantities of the order P p 2 t P and p being any two points in the 

 element ; that this is the meaning is shown by the fact that 

 the whole investigation depends on quantities of the order Pj9. 

 Now, not only in the case where udx + v d y + w d z is an 

 exact differential, but also in the case where it is integrable 

 by a factor, there exists a surface of displacement passing 

 through P, and the above statement will be true for an ele- 

 ment of this surface. But it will not generally be true for an 

 element of three dimensions; for, let p be taken in the line of 

 direction passing through P ; then, if «x be the radius of ab- 

 solute curvature of this line at the point P, and Pp = 8 s, 

 the angle between the tangents at P and p will be ultimately 



8 s 



— . Neglecting quantities of the order 8 s 9 , a line PT' drawn 

 w 



through P parallel to the tangent at P may be taken instead 



of the tangent at p. Now, even if we suppose the line P T' 



to pass through the focal line which is at a distance r from 



p, the least distance between it and the other focal line, which 



8s 

 is at a distance r 1 from p, will be ultimately r' — . Hence, 



it cannot ultimately pass through both focal lines, unless "ro- 

 be at every point infinite, i. e. unless all the lines of motion be 

 right lines, which is evidently a very limited case. Conse- 

 quently, it is only in this case that it is proved that surfaces of 

 displacement are surfaces of equal velocity. 



There is another part of Professor Challis's reasoning with 



which I cannot agree. It is that d (-rr) or , , , dx 

 ■ \dt J dtdx 



d*$ 7 d?4> , L . . c 



+ — — ►*- dy + , , d n — 0, in passing from one point to 



another of a surface of displacement. For, d ( — \ m 



the differential equation to a family of surfaces whose general 



equation is — j— = C, which family of surfaces is in general 



quite different from that whose equation is <p = 0. Now the 

 proof requires that the variations dx> dy, dz should be taken 

 along that surface of the second family which passes through 



the point {x,y, z), whereas the variations for which d( — ) = 0, 



must be taken along that surface of the first family which 

 passes through the same point. If <p = r[/ (t) (.r 2 — j/ 2 )+% {t)xy 9 

 for instance, these two surfaces will be different. 



is 



