PROFESSOR CAYLEY ON THE SEXTACTIC POINTS OP A PLANE CURVE. 549 
Article Nos. 7 & 8 . — Notations and Remarks. 
7. Writing, as in my former memoir, A, B, C for the first differential coefficients of U, 
we have Bv— C^, Gx — Av, Ay — BX for the values of dx, dy , dz, and instead of the 
symbol B used in my former memoir, I use indifferently the original symbol B 15 or write 
instead thereof B, to denote the resulting value 
3 i (=^)=(Bv-C^)B,+(CX-A»)^+(A|m ( -Bx)B > , 
and I remark here that for any function whatever O, we have 
BO= 
A , B , C 
X , y , v 
B,Q, B.Q, B.O 
=Jac. (U, a, Q), 
where §=Xx-\-yy-{-vz. I write, as in the former memoir, 
®=(a, as, c, f, <g, p, »)*; 
and also 
V=(S3, 3S, C, S': <&, y j : dy, B,), 
which new symbol V serves to express the functions IT, □ , occurring in the former 
memoir; viz. we have n=2VO, □=2VH, so that the symbols 14, □ are not any 
longer required. 
8. I remark that the symbols B, V are each of them a linear function of (d*, B y , BJ, 
with coefficients which are functions of the variables (x, y, z) ; and this being so, that 
for any function 14 whatever, we have 
B(vn)=(B.v)n+Bvn, 
viz. in B(VI1) we operate with V on 14, thereby obtaining VII, and then with B on VII; 
in (B.V)II we operate with B upon V in so far as V is a function of (x, y, z), thus 
obtaining a new operating symbol B.V, a linear function of (B^,, B^, B*), and then 
operate with B.V upon II; and lastly, in BVII, we simply multiply together B and V, 
thus obtaining a new operating symbol BV of the form (B a ,, B^, B„)' 2 , and then operate 
therewith on II ; it is clear that, as regards the last-mentioned mode of combination, the 
symbols B and V are convertible, or BV = VB, that is, BVn = VdII. 
It is to be observed throughout the memoir that the point ( . ) is used (as above 
in B.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 BV) denotes that the 
symbols of operation are simply multiplied together; and when BV 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 BVII, the resulting operation B V is performed upon II 
as operand ; if instead of the single letter II we have a compound symbol such as HU 
or HV^, so that the expression is BHU, BHV&, BVHU or BVHV3-, then it is to be 
understood that it is merely the immediately following function H which is operated 
upon by B or BV ; in the few instances where any ambiguity might arise a special 
explanation is given. 
