1202 
MR. R. F. GWYTHER OK THE DIPFEREKTIAL 
The first of these equations is satisfied by 
Pr-A 1 . . . , 
2 P, + 3P3-10PiP3+14P,= J 
(P.^ - P^MaPi + 8 P 3 - lOPiP^ + 14PiV.(44), 
and 
Hence 
(43). 
is the function of differential invariants which gives a first integral of the differential 
equation, or in other words, is the absolute invariant of the cubic. 
24. The general form of the matrix of a cubic. 
In all cases the matrix is a function of iq, Lq, and the differential invariants. In 
the most general case the matrix of the cubic will contain uf but if we take the 
cubic to touch the standard curve (which is actually to be the curve itself), the 
matrix would contain only uf. 
The coefficient of if will also be simply a differential invariant, and, from the 
nature of the operators and Xlg, the complete form is found to be 
xfj + xjjfjQ -j- “h (^ + ^ 1^6 + ^ 31 ^ 6 ^) 
with the condition which will be presently seen 
f + “1" 
Taking the point of contact with the standard curve as temporary origin, and 
vniting the equation 
hiT^ + + Sa.^Trf -j- + tomir^ + 3ag^“ + Sc^tt = 0, 
we shall have 
h = matrix, as above 
361 = — w/w.g -f- 4 i//^Lg® 4 - {(f)^ + 243Lg)%,} 
+ [^ + + 2/^4] Cfcg 
Sug = 11-2^11-^ "h d“ 
+ 'if (<^ + 4*1^6 + 4*2J-‘g^ + 2/%) — 'if'iiG { 4^1 + 2<^3Lg) cfg + uffa^~ 
a = — u.hf (i/ig + 4 i// 4 Lq) — ufu-^ ((/)i + 243Lg) + 'if ( 2 / + «3 
with the condition 
/+ 'a 5^ 4* 2 + if 4^4: = 0- 
363 = Wg® ((/) + </>iLg 4 " <^oLg^ 4" 2 jdq) 
(hn = ifu^ {(f)^ + 2(/)gLe) - 2iffa.^ 
Sttg = — Uff(f)2 — luff 
,Sc, — ?rd/‘ 
(45). 
Also 
