71 
results would be three equations between each pair of the variables, 
and these equations would be of the mn” degree, (See note on 
(602,) Geometry.) Hence the curve sought is of the mn™ degree. 
Prop. 
(5) Three right lines intersecting at the same point, intersect an 
algebraic surface ; to determine the relation between the con- 
tinued product of the intercepts of each between their common 
intersection and the surface. 
Let these right lines themselves be assumed as axes of coordinates. 
In order to determine their intercepts between the origin and the 
surface let each pair or variables be successively suppose = 0. 
The result will be three equations of the forms 
Age Be) or wwe. Me EN =O. 
vy + By + yw wy MY NO, 
Ale + Bs" + c’2"* . . M2 +N. 
The continued products of the roots of these equations are res- 
pectively NX, %, %; hence the ratio of each pair of these pro- 
A A A 
ducts is the reciprocal ratio of the coefficients of the highest powers 
of the variables respectively which enter the equations whose roots 
are their factors. As the transformation of origin does not affect 
the values of a, 4’, a”, these ratios remain the same for all parallel 
systems of axes. Therefore if three right lines parallel to three 
right lines given in position and intersecting at the same point in- 
tersect an algebraic surface of the n™ order, the continued product 
