1184 
MR. R. F. GWYTHER ON THE DIFFERENTIAL 
(I - X) {(f - *) 3^- + (ri-y) 3^ + (^-p) 
-df 
0/7 
= {ftl -^71 - 2) ^ 5 
{i-y)\{^- + {i-~y) + (^-p) e7^'_iA “ 
s/ 
3(1- .'c) 
{v — y) 
{TT-f) 
9/ 
— S/«S;; {tYI 1 Tl ~~ 2^ — \ ^ ’ 
S/ 
S/ 
3/ 
(^ _ p) j (f _ .X) 3 -— + (,-;/) 3^^^ + (^ - j.) - rf/ 
(v — y) 
(tt —_p) 
- Sj,;;S;; SyjS^ ( “j“ (/ l) ^^ — P1 ’ 
0('m. 11 
and 
(tt 
i^) 
cc) 
• y) 
3/ 
3 (tt — p) 
3/ 
d(^ -x) 
3/ 
d{rj - y) 
{d + c4)/— S;„S„ ^ > 
3/ 
— f ~\~ 7Yl 
— “ 1 “ ^ ^/«, n 
3c ’ 
3c 
n 
The coefficient of the highest power of n — j) is not only isobaric in x and in y, but 
is homoiobaric, of weight to, in x and y. Also d. denotes the degree of that coeffi¬ 
cient, d denoting the algebraic degree of f. 
We have, as in the case of plane curves, a theory of deduction from a matrix, and 
a theory of eduction (in the form of a Jacobian function) of both matrices and 
invariants. 
The subject is quite similar at all steps to the investigation for plane curves in this 
paper, and offers little additional interest except in the solution of the differential 
equations which the matrices and invariants satisfy. By solution wm can verify 
Halphen’s statement in the last j)aragraph of his thesis (p. 60). 
There is an invariant of the second and of the third order, two of the fourth and 
six of the fifth. There are three matrices, not invariants, of the fourth order, and a 
system of invariantal coordinates (§ 5 below) can be formed. 
The theory of covariants of twisted curves is sufficiently different to merit a 
separate investigation, but the germ of it already exists in Halphen’s paper (quoted 
above) “ On the Invariants of Twisted Curves.” 
The conditions for covariants of surfaces given above, include those previously 
obtained by Mr. E. B. Elliott, “ On Ternary and ?i-ary Reciprocants,” ‘ Proc, 
Math. Soc.,’ No. 262, 1886 .—August 2, 1893.] 
