88 
PROFESSOR CAYLEY ON THE CURVES 
(viz. = 1 2 (os/3 ' + oj'/ 3) -f- 4/3/3', &c.), and these values give identically 
2^" — Sv" + 3/ — 2<r" = 0, 
which is the foregoing equation. And I assume that the theorem extends to the case 
of two inseparable conditions 2X, hut in this case I do not even know where the proof 
is to be sought for. 
The characteristics ((/l, v', g') of the three conditions 3X are in general independent. 
36. It has been mentioned that if (a, (3) are the representatives of the condition X, 
and (p, v) the characteristics of the conditions 4Z, then 
(X, 4Z)=«H-0r; 
this is the most convenient form of the theorem, hut as (a, (3) are known functions of 
the characteristics (fjil", v 1 ", g'", o’", r") of the condition X, the equation is in effect an ex- 
pression for (X, 4Z) in terms of the characteristics of X and 4Z respectively. 
There is, similarly, an expression for (2X, 3Z) in terms of the characteristics 
(/!//, v', g', a') of 3Z (satisfying the relation (j ! — fV -\-%g'—<j:'=0) and the characteristics 
(/-&, v, g) of 2X, viz. we have 
(2X, 3Z)= K - 
* +" (— Ip'+tV+tW’— M 
+*'( V-i' )• 
This may he easily proved in the case where the conditions 2X are two separable condi- 
tions X, X' having the representatives (a, (3), (a', (3') respectively, and the conditions 3Z 
three separable conditions Z ", 71", 71" having the representatives (a", (3"), (a,'")(j3'"), (a!'", (3"") 
respectively ; we have, in fact, 
**'=(!, 2, 4X«, /3'), ^=(1, 2, 4, 4J a ", (3")( a "', (3 (3""), 
" f =(2, 4, 41 „ „ ), v=(2, 4, 4, 2X „ „ „ ), 
g'=(4, 4, 2X „ ,, ), g=(4, 4, 2, IX „ „ „ ), 
^=(4, 2, IX „ „ ); 
and with these values the function 
(X, X', 7", 71", 71'"), =(1, 2, 4, 4, 2, 1X«, j3)(«f, j3')(«", 0 (3”')(*"", (3"") 
is found to be expressible as above in terms of (^, v, g), (j ul, 1 , g, <r') ; but I do not know 
how to conduct the proof for the inseparable conditions 2X and 3Z. 
37. It may be remarked by way of verification that writing successively 
(3Z)= (.•.), (:/), (■//), (///), 
(f*, g)=(l, 2, 4), (2, 4, 4), (4, 4, 2), (4, 4, 1), 
that is, 
