( 814 ) 
Abt Eke Ce C,) v 
B= — (be) dep DS (ae 
N (“) = (un =) —N, , p(<) — P, ete. 
Y eu tev Y 
where N, P, ete. are homogeneous functions of w' and v' of degree zero. 
Hence, if we arrange according to the degrees of w and v' the 
numerator takes the form 
(edt [NB 42a By ed 
ris (5, Cs) Vi LR, By he S, veal 
+ ela, ¢) v [2 N, 8 + Py] 
— eb, ¢) R, vy 
Hee [Rey + Sy") 
+ Q° (a, Cy) N, v 
+ ete By 
and in the same way the denominator may be written 
(a, ¢,) u LN, B + P, By + Q, v7] 
— (b, ¢,) v [R, By +S, 77] 
Hela, ¢) uv [2M, 8 + P, 1] 
— @ (b: ¢,) Ry wy ij 
— ec [R, By + 8,1] 
+ 9? (ae) Nou 
—o’c, R, y. 
If we examine these values it is evident that the equation (3) reduces to 
dv! K,+-M,+v(N,4+¢) a 
du xA,+L,+u'(N,+0¢) 
where c represents a constant, H, and H, homogeneous functions 
of the first degree and L, M, N, homogeneous functions of the 
second degree. 
From the values 
EN ir ete, Ry 
K,= eey 
ALE o'(a,e) N, 
we may readily induce that if, in (1) A(z) is absent H, and K, 
must be zero and if in (1) N(e) is absent, we have c= 0. 
The preceding considerations furnish the inference that every 
homographic substitution applied to an equation (1), followed by a 
transformation to parallel axes through the point a= y =O gives 
necessarily an equation of the form (4). 
