276 Differential Covariants of Twisted Curves, &c. [June 21, 



<"& tt '2? _ / Qfi\ _ O 



- ~ 2 - 



- _ _i_ -T 



5 T> 2 / ' 



We can now write down the leading terms in the expansions of 

 a a and /3, t , and we can differentiate the expressions, remembering that 

 after differentiation we put a t = 6 4 = . . . . = a 6 = a 6 , 07 = a 7 , &c. 



All the expressions I, J, K, &c., vanish if not differentiated, and all 

 the part contained in R vanishes after differentiation as well. We 

 have then only to consider that part of the differential coefficient of 

 I, J, K, &c., which does not vanish. Writing this [I'], &c., we get 



[J']=12,, [T']=.6 6 , 



[K']=0, 



and if ['] stands for the invariantive part of the differential co- 

 efficient of , being the only part when referred to the canonical axes, 

 we have 



-3)* n _ 2 + ---- + (nm 1) ft m * H -m+ ---- 

 2)* n _ 2 + .... +(n m) m ^.i<x n - m + .... + 



/J f 



7 < (w 4)a n ^! .... -2(n m l) j3 m +ix n - m . 



6 L 



and 



_ 2 



12 a6 {(?t-3)/3_ 2 + ---- +(w-m 



(n 2) #,_,+ ---- + (n-m) *m 



7 {(n 4) yS H -i- ---- -2 O-m- 



This shows the mode by which the general values of these differ- 

 ential coefficients are found. The mode by which the series of 

 canonical invariants can be consecutively determined follows, and the 

 relations connecting the canonical invariants of a curve with those of 



