447] ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 221 
viz., one of the quartics passes through B"', O'"; through D'", E”\ F'" each once; and 
through G"\ H'", I'" each twice. And so in the fifth figure the Jacobian consists of 
3 octics B""C"" (D""E""F""\ (G""H"'T"\ (J""KL""\, 
3 quartics J)""E"" 
3 conics G""H"" (J""K""L""\, 
3 lines 
and so on. 
66. The conditions are in each case sufficient for the determination of the curve. 
This depends on the numerical relation 
4 + 3 {1.2 + 2.3 + 4.5 + 8.9 ... + 2 0 ( 2 0 + 1)} = 2 0+1 (2 0+1 + 3). 
The term in { } is 
1 + 4 + 16 ... + 2 20 
+ 1 + 2 + 4 ... + 2 e , 
that is 
226+2 _ l 96+1 - 1 
2 2 — 1 2 — 1 ’ 
which is 
= £ [2 20+2 —1 + 3 (2 e+1 — 1)], 
= 4 [2 20+2 + 3.2 0+1 - 4]; 
and the relation is thus identically true. 
67. Conversely, in the transformation between the first figure and the several 
other figures respectively, the Jacobian of the first figure is 
3 lines 
AB ; and so 
3 conics DE {ABG\ 
3 lines AB 
3 quartics GH (.DEF\ (ABG\ 
3 conics DE (ABC), 
3 lines AB 
3 octics JK (GHI), (DEF), ( ABC% 
for order 2, between first and second figures; 
for order 4, between first and third figures; 
>■ for order 8, between first and fourth figures; 
3 quartics 
3 conics 
3 lines 
and so on. 
GH (DEF), (ABC) s 
DE (ABC), 
AB 
for order 16, between first and fifth figures;