1890.] A. Mukhopadhyay — On a Curve of Aberrancy. CI 



Therefore, putting 



dy dz = IdS, dx dz = mdS, dx dy = ndS, 

 where I, m, n are the direction cosines of the normal, we have 



/ 



{lidx + vdz -\r wdz) 



/fl/n rl'ii\ \ 



d8 



~ I / ( \% dz) \dz dx) \dx dy) ) 

 = 2 j C {U ^ mrj -V nV) dS, 



which is Stokes's Theorem. It is worth noting that as no physical con- 

 ception enters into the above proof, it holds good whether we regard the 

 theorem as a purely analytical one or as merely furnishing a formula for 

 hydrokinetic circulation. 



V. — On a Curve of Aberrancy. 

 By AsuTOSH MuKHOPADHTAT, M. A., F. R. A. S., F. R. S. E. 



[Received May 23rd ;— Read June 5th, 1889.] 



If a curve be referred to rectangular axes drawn through any 

 origin, the coordinates (a, /?) of the centre of aberrancy, which is the 

 centre of the escalating conic at any given point (a?, ?/) of the curve, 

 are given in the most general form by the system 



_. Zqr 



^ ^ Sqs - 5r2 



where p, q, r, s are the successive differential coefficients of y with 

 respect to x.* The locus of (a, /3) is called the aberrancy curve of the 

 given curve, and in this note, I shall investigate the aberrancy curve 

 of a plane cubic of Newton's fourth classt 



y = ax"^ + Sbx^ 4- Sex 4- d 

 in which the diametral conic degenerates into the line at infinity. 

 We have 



p = 3 (ax^ + 2bx + c) 

 g' = 6 (ax + b) 

 r = 6a 

 s= 



* J. A. S. B. 1888, vol. Ivii, part ii, p. 324. 



t Salmon's Higher Plane Curves, (Ed. 1879), p. 177. 



