272 Mr. R. F. Gwyther. The Differential [June 21, 



becomes now slightly distorted, so that the locus of the most probable 

 y deviation, corresponding to an assigned x deviation is no longer a 

 straight line, but a parabola. 



X. " The Differential Covariants of Twisted Curves, with some 

 Illustrations of the Application to Quartic Curves." By 

 R. F. GWYTHER, M.A., Fielden Lecturer in Mathematics, 

 Owens College, Manchester. Communicated by Professor 

 HORACE LAMB, F.R.S. Received May 31, 1894. 



(Abstract.) 



The object of the earlier parts of this paper is to obtain relations 

 connecting what Halphen* calls the canonical invariants of the curve, 

 without the intervention of what are called by him the fundamental 

 invariants. In any geometrical investigation it is the canonical in- 

 variants which present themselves, and the relations between the con- 

 secutive canonical invariants, and the values of their differential 

 coefficients must, if the method of investigation is to be used, be 

 expressed, in terms of canonical invariants only, without the inter- 

 vention of the other series of invariants which Halphen treats as 

 fundamental. 



In the paper, the notation of Halphen's paper cited above is gener- 

 ally followed, but the mode of initial investigation, as in a previous 

 paper on covariants of plane curves,f is made to depend upon a 

 homographic transformation with infinitesimal arguments. 



Writing , ij, for the coordinates of a current point, #, y, z for the 

 coordinates of a point on a standard curve, and y n and z n for 

 ctPyjdaf . n ! and d"z/dx n . n !, the arguments of a covariant function are 

 shown to be 



g = zz^yyi ? 



with a n = y^nynZ 2 jy 2 z 3 y 3 z 2 , 



while y z z 3 2/3^2 is an invariant which will not appear independently. 



The other conditions that a function (/, gr, h, a n , 6. . . . ) may be 

 a covariant function are (1) that it shall be isobaric, counting/, g, h, a n , 



* " Sur les Invariants Difflirentiels des Courbes Gauches," ' Journal de 1'^cole 

 1'olytechnique,' cahier 47, vol. 28, 1880. 



t ' Phil. Trans.,' vol. 184 (A), 1893, p. 1171. 



