160 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
which determines the coefficients in (4). (*) Putting w,, for 
1 é” Tie rs v+l 
rl, al, de Oy? We have yy gr * ¥ (atu) = 2 ((%) ty) +2 ((%) ty 
+ (%) Uv) + & ces (CO) Uy + Oy) thy + (2) Mv) + &e., and this in 
(1’) gives an equation of the form 
0=2A,+21A,+27A,+ 2A, + Ke. (8) 
viz: 0 = 2° [(0,) Mo} + 2 [(09) to + (01) Moo + (Lo) Mor] 
+ 2° [ (0p) Ugo + (01) Myo + (Oz) Moo Ao) Mar + Ai) Mort (2) Mop] 
4° [(0p) Usp + (01) May + (02) Myo + (95) Moo A Lo) Mort Ay) Mh 
+ Ag) tor HF (20) he + C21) Moz + (8p) Mos] + Ke. (8’) 
for the abscissae of points common to (1) and (3). Similarly for 
the abscissae of points common to (2) and (3) we get an equation 
of the form 
0O=2 Bote B, +7 B,+2 B, + &e. (9) 
viz: 0 = 2° [(O4) % J + 2 (00g) Mo + (1) Yo + Go) Mm] + &e. 8’) 
Let (2) contain at least p parameters, enabling us to pass (2) 
through p of the intersections of (1) with (8). When this is done 
we have the equation 0 = 2° (A,— B,) +. 2' (A, — B,) +2’ (A,—B,) 
+ &c. (10) true for the p values of x corresponding to the p points 
common to (1), (2), (3). Let the » common points move to the 
origin, (10) must have p roots equal zero, that is, 0 = A, — 
B, 0= A, =— By, 0 = A, — B, . 2.0 = A,_, — Bo eee 
If we suppose (3) the parabolic representative of (1), 2 in (8) 
becomes indeterminate, and hence besides 0 = A, we have also 
C= A 0= A, Ge. 
that is, 0 ng with 
dy 6 
[ = t+ Zt5, 
| va Ley a FS «Ae (ay ee 
2.08" deatin Bide on, 2 NOs) On 
Oh ck AG Od 1d’ Of 1 dyof 
= 327 QaziP + Nae de, 1 Baeo, G2) 
1 (2) 5 2 Bae ii Feet H(z oe 
| boa \ag) ago TBI de dB an? TBI Nae) oy 
L  &e. &e. _ &e. 
* Putting x = 1 in (3/7) and (4), we obtain the multinomial theorem in the form 
(204 -+ Wy + Ws + &Ke.)¥ = (%) + (%) +e (%) + Ke. 
