PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
527 
I assume that (x, y, z) are auxiliars, reserving for the concluding articles of the present 
memoir the considerations which sustain this assumption. 
275. The separation into regions is effected as follows: — We have identically (see ante , 
No. 255) 
1 6P = JK 4 + 8LK 3 - 2 J 2 LK 2 - 72 JL 2 K - 4 32L 3 + J 3 L 2 , 
or putting for K its value = x ^g{J 2 — D), this is 
2 32 I 2 = J(J 2 -D) 4 H-&c. 
= (J 3 — 2“L) 2 (J 3 — 3 3 .2 10 L) 
+D J( - 4J 6 + 61 . 2 10 J 3 L+ 1 44 . 2 20 L 2 ) 
+ D 2 J 2 ( 6J 3 — 2 ,0 .29L) 
+D 3 ( — 4J 3 — 2 10 L) 
+D 4 J. 
Or writing as above 
whence also 
this is 
2 n L-J 3 
J 3 ’ 
2 U L 
D 
'J 2 ’ 
l+x= 
2 M j§=— tf 2 {f(l H-tf)— 1} 
+y {36(1+^) 2 — ^(1+^’)— 4} 
-3/ 2 {¥(i+^)-6} 
+y\ 
or, what is the same thing, 
2.2 32 ^= -3# 3 -# 2 
-\-y (72a? 2 -f 205# -1-125) 
+f(- 29a?- 17) 
+ f(-x-9) 
+*/ 4 -2 
=<p(x, y) suppose. 
276. Hence also writing z— J, we have 
z<p(x, y)= 2.2 32 j- 8 = + 
or the equation of the bounding surface may be taken to be 
z<p(x, y) = 0, 
that is, the bounding surface is composed of the plane 2=0, and the cylinder <p(x, y) = 0. 
Taking the plane of the paper for the plane 2=0, the cylinder meets this plane in a 
