447] ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
229 
Next, the case n = 2, m 1 = 2, Aj = 0, a x = 1 : the reciprocal transformation is of the 
order ri = 2 ; it is evidently not of the form above considered (for this would make 
the original transformation to be of the order 3). Hence, assuming (as it seems 
allowable to do) that the principal system does not contain any multiple point or 
curve, the reciprocal transformation will be of the same form as the original one ; 
viz., we shall have ri = 2, mi = 2, А/ = 0, a/ = 1. 
87. Considering next the cubic transformations, or those belonging to w = 3; in 
the case m x — 4, h x = 4, a 1 = 2, the reciprocal transformation is of the order 9 — 4, = 5 ; 
and in the case m x = 5, h x = 5, a x = 1, the reciprocal transformation is of the order 
9 — 5, = 4 : I do not consider these cases. But m x = 6, h x = 7, a x = 0, the reciprocal 
transformation is of the order 9 — 6, = 3 ; and assuming (as seems allowable) that the 
principal system does not contain any multiple point or curve, it must be of the 
same form as the original transformation, that is, we must have ri = 3, те/ = 6, hi = 7, 
ai = 0. 
88. The transformations to be studied are thus,—1° The quadri-quadric trans 
formation n—2, m x = 2, A x = 0, oq = 1, and ri — 2, mi = 2, А/= 0, a/ = 1 ; the principal 
system in each space consists of a point and of a conic (which may be a pair of 
intersecting lines) ; and the surfaces are quadrics. 2° The quadri-cubic transformation 
n = 2, m x — 1, Aj = 0, = and ri = 3, a/ = 0, mi = 3, A/ = 3, mi = 1, A 2 ' = 0: in the first 
space the principal system consists of three points and a line, and the surfaces are 
quadrics : in the second figure, the principal system consists of three simple lines and a 
double line ; and the surfaces are cubic surfaces passing through this principal system, 
that is, they are cubic scrolls. 3° The cubo-cubic transformation n — 3, a x = 0, m x = 6, 
h x = 7, and ri = 3, а/ = 0, те/ = 6, hi = 7 ; in each space the principal system is a sextic 
curve with seven apparent double points (but there are different cases to be considered 
according as the sextic curve does or does not break up into inferior curves), and 
the surfaces are cubic surfaces through the sextic curve. 
The Quadri-quadric Transformation between Two Spaces. 
89. Starting from the equations x' : y : ri : w = X : Y : Z : W, we have here 
X = 0, &c., quadric surfaces passing through a given point and a given conic (which 
may be a pair of intersecting lines). Take x = 0, y = 0, z=0 for the coordinates of 
the given point; w— 0 for the equation of the plane of the conic; the conic is then 
given as the intersection of this plane by a cone having the given point for its 
vertex ; or say the equations of the conic are w = 0, {a,...Tfc, y, z) 2 — 0; the general 
equation of a quadric through the point and conic is w (ax + /3y + 7z) + 8 (a,. ..^x, y, z) 2 = 0; 
and it hence appears that the equations of the transformation may be taken to be 
x' : y' : z' : w' = xw : yw : zw : (a, ...$#, y, z) 2 \ 
these give at once a reciprocal system of the same form ; viz., the two sets are 
x' : y' : z’ : w =xw : yw : zw : (a, ...$#, у, z) 2 , 
: у : z : w — xw' : y'w' : z'w : (a, ... у, z') 2 . 
and 
x 
z 
w — xw