1890.] A. Mukhopadhyay — On a Curve of Aberrancy. 61 



Therefore, putting 



dy dz = Id 8, dx dz = mdS, dx dy = ndS, 

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



If 



I (udx + vdz + wdz) 



i /div dv\ /du dw\ /dv du\ \ 



[ \dy dz) \dz dx) \dx dy) ) 



= 2 | | (# + my + nQ cZ^, 



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 Mukhopadhyay, M. A., F. R. A. S., F. R. S. E. 



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



If a curve be referred to rectangular axes drawn through any 

 origin, the coordinates (a, /3) of the centre of aberrancy, which is the 

 centre of the osculating conic at any given point (x, y) of the curve 

 are given in the most general form by the system 



Sqr 



a = x — 



P = V - 



3qs — 5r 8 

 3q (pr - 3<7 2 ) 



3qs — 5r a 



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 ■=■ a* 3 + 2>hx % + 3cx + d 

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

 We have 



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

 q = 6 {ax + b) 

 r = 6a 

 s= 



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



f Salmon's Higher Plane Carves, (Ed. 1879), p. 177. 



