226 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
[341 
and also 
V = (31, S3, <£, & @, fi, v\d x , dy, d z ), 
which new symbol V serves to express the functions II, □, occurring in the former 
memoir; viz. we have II = 2V<i>, □ = 2VfT, so that the symbols II, □ are not any 
longer required. 
8. I remark that the symbols 9, V are each of them a linear function of 
(9a;, d y , d z ), with coefficients which are functions of the variables (x, y, z), and this 
being so, that for any function II whatever, we have 
9(VII) = (9.V) II +9VII, 
viz. in 9(VII) we operate with V on II, thereby obtaining VII, and then with 9 on 
VII; in (9.V) II we operate with 9 upon V in so far as V is a function of 
(x, y, z), thus obtaining a new operating symbol 9.V, a linear function of (d x , d y , d z ), 
and then operate with 9.V upon II; and lastly, in 9VII, we simply multiply 
together 9 and V, thus obtaining a new operating symbol 9 V of the form (9*, d y , 9 Z ) 2 , 
and then operate therewith on II; it is clear that, as regards the last-mentioned 
mode of combination, the symbols 9 and V are convertible, or 9V = V9, that is, 
dVII= V9II. 
It is to be observed throughout the memoir that the point (.) is used (as above 
in 9.V) when an operation is performed upon a symbol of operation as operand; the 
mere apposition of two or more symbols of operation (as above in 9V) denotes that 
the symbols of operation are simply multiplied together; and when 9V is followed by 
a letter II denoting not a symbol of operation, but a mere function of the coordinates, 
that is in an expression such as 9 VII, the resulting operation 9 V is performed 
upon II as operand; if instead of the single letter II we have a compound symbol 
such as HTJ or so that the expression is dHTJ, 9HV^f, 9VHU or 9V_ff'V&, 
then it is to be understood that it is merely the immediately following function H 
which is operated upon by 9 or 9V ; in the few instances where any ambiguity 
might arise a special explanation is given. 
Article Nos. 9 to 11.—First transformation. 
9. We have, assuming always U = 0, the following formulae (see post, Article 
Nos. 31 to 33): 
(9j 5 + 1(V9 8 + 109^3 + 159A 2 ) U 
= !<27m> - 96m + 81) H№ + (17m 2 - 56m + 51) <J>3/f] 
+ (S^iy K- Um ~ 22)(0.'V) H - (10m - 18) d V H]