420 



Mr. R. F. Gwyther. On the Differential [May 4, 



straight muscle by similar regions of the cortex of the same side as 

 the muscle. A point of difference — and it is a suggestive one — be- 

 tween the two cases appears to be that under inhibitory relaxation of 

 the rectus externus from cortical excitation the globus may rotate 

 beyond the middle line of the palpebral fissure, whereas under 

 cortical relaxation of the rectus internus the eyeball may travel up to 

 that middle line, but very rarely, if ever, trespasses beyond it. 



It thus seems clear that by experiment abundant support can be 

 obtained for the supposition put forward by Charles Bell. 



V. 66 On the Differential Covariants of Plane Curves, and the 

 Operators employed in their Development." By R. F. 

 Gwyther, M..A., Fielden Lecturer in Mathematics, Owens 

 College, Manchester. Communicated by Professor HORACE 

 Lamb, F.R.S. Received April 14, 1893. 



(Abstract.) 



Consider any point (x f y) on a standard plane curve, and write 



C&i, 



&c, for dy/dx, d*yjdx 2 . 2 !, d 3 y/dx 3 . 3 !, &c, the differential co- 

 efficients being derived from the equation to the curve. Let (^.7) 

 be the current coordinates of a point which moves so that/^, rj, x, y, 

 01, a 2 , a 3 ,. . . . ) = 0, say, on a trajectory of the standard curve. Now 

 let a general nomographic transformation affect r/, and x, y alike ; 

 the function which replaces rj, x, y, a Xi . . . . ) will generally entirely 

 change its character; if, however, it retains the same form (except 

 that it is affected by a certain factor of which the form is to be 

 found), I define it to be a differential covariant of the standard 

 curve. 



It is obvious that among the covariants will be found tangents and 

 polar curves, and the ordinary covariant curves. It is proposed to 

 investigate the subject generally, and to find the relations of the 

 covariant with the contra variant curves. 



§i- 



For the purpose of obtaining the forms of the linear partial differ- 

 ential equations which express the conditions that a function should 

 be a covariant function, an infinitesimal homographic transformation 

 is employed, expressed by the relations 



g _ y_ 1 



X + B^ - Y " AX + BY + l' 



where A, B, and Bi are small, with identical relations for £ and rj. 

 The conditions found may be stated as follows. 



