274 Mr. R. F. Gwyther. The Differential [June 21 r 



where 4 = 4 , 



u s = fl s 3& 4 a 4 2 , 



and 



The planes, whose equations are = 0, y = 0, % = 0, ia = 0, form 

 a tetrahedron, which will be called the canonical tetrahedron of 

 reference at any point on the curve, and if we put 



the coefficients of the expansions of ?/ and z in terms of are the 

 canonical invariants. 



We write these expansions 



y = x* +ft 1 x'+&c., 



Z = X? + eti,X* + aiiX 1 +...., 



and since g = / 2 + .... +b i f i +&c., 



we see that with respect to the canonical axes, 



a 4 = 6 4 = a 5 = 6 6 = J 6 = 0. 



To obtain the result of differentiating a canonical invariant, we must- 

 begin with a general change of axes, and after differentiation suppose 

 that these are the canonical axes, and put a 4 = J 4 = &c., = 0, a& 

 above. For this purpose, I show that if 



Y ^ v ^ -7 



-A. , I , Zi 



w w u 



denote any homographic transformation, we shall have 



A, = 



&c. + &c. 



