114 
MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL IRA^ARIANTS. 
which are combinations of polar emanants of Aq and A'q, the fundamental quadratic 
forms. And, from the note to § 24, it follows that they can also be represented in the 
forms 
3 34 + 2-’ S + ’’ 3; + ^ 5 + ' 
dp 
0 0 0 0 9\a/ 
and also in the forms 
i 0^/ + ’’^ 
dp 
dr- 
dr 
J//0, 
2 yJ- dq + A. + ^ ^ ^ 
0.S' 
ds 
0f 
c 
dt 
0> 
0- 
36. Returning now to the functional invariants of only a single dependent variable, 
we have seen that they are combinations of the simultaneous covariants of Aq, A^, 
A 2 , . . . , considered as binary forms in q and — p ; and all these simultaneous 
covariants satisfy the equations — 0, and must, therefore, be expressible in terms 
of iq, , U-, . . . , u^, .... The actual expressions may be obtained as follows:— 
From the values of the quantities u we have 
< Ut^s = Wg + qyih, 
_ = tq + ^qnu + qru. ; 
r 
d = zq. 
j ^qc = - n.Q A pW5> 
?q2 0 = tq — 2piq + qru-^, 
ii^a = — Ug + dqmri — dpru^ + ; 
l - Wg, 
u^h = — iqo + p^^g, 
\ = Wn - 2piqo + p%.g, 
— SjAiqo + _p®?q, 
= ^qg — 4prq3 + 6^Aiq^ — 4p^iqQ + phqQ ; 
and so on. It thus appears that any differential coefficient of 2 , when multiplied by 
a power of ^q equal to the x-grade of the differential coefficient, is linearly expressible 
in terms of the quantities u proper to its rank, the coefficients of these quantities u 
in the expression being powers of u. 
But in the case of the function A,;_ 3 , which is 
0’ ^"-1, n p)"^’ 
