PROFESSOR A. R. FORSYTH ON THE DIFFERENTIAL INVARIANTS 
ol»6 
and the value of the latter expression is 
2V- 
pr _ 1, r/B 1 r/B 
p'J fJs t' dn 
In this way the actual values of a large number of the Invariants belonging to the 
asyzygetic aggregate can be obtained. The asyzygetic aggregate of two cubics is 
known. The asyzygetic aggregate, arising when a quadratic is associated with 
a system asyzygetically conqdete in itself, is also known; so that the asyzygetic 
aggregate belonging to a’b, V'o^, idn can be obtained by the application of 
known theorems. 
Further, the asyzygetic aggregate of a cubic and a quartic is known, so that 
tlie asyzygetic aggregate could l)e obtained f(.)r aq, a'b, aq, aq, and also for 
'aq, tab, ia'g, u\. But, so far as I am aware, the asyzygetic aggregate of either 
two cubics and one quartic, or a cubic and any system asyzygetically complete in 
itself, is not known ; as soon as either is known, the results could be applied to 
obtain the asyzygetic aggregate for iv. 2 , iv'o, ly'b, w\, aq, that is, the complete 
system of concomitants in terms of which any rational integral invariant can be 
expressed as a rational integral function. 
The Geometrical Magnitudes which are Independent. 
55. As regards the quantities, which have served to assign the geometrical 
significance of the several invariants, some inferences can be drawn from the results 
obtained. Denoting by ;)( the angle between the curve and the line of curvature 
connected with pj, we have 
cos' X _p sill- X , 
Pi Pi 
]_ 
Ti 
I 
cos y Sin y 
1 
-\~ 
P\ P: 
K = 
PiPi 
so that not more than three of the ([uantities --, , H, K are independent. For 
P ^ 
puiqioses of expression, we have retained H. 
P 
B and ~ • 
P 
There are also the quantities 
To the order of derivatives which occur in the invariants that have been 
constructed, the geometric magnitudes, which might be expected to occur in the 
values of the invariants, are as follows :— 
