360 ME. W. SPOTTISWOODE ON HTPEE J ACOBI AN SURFACES AND CURVES. 
Again, differentiating the right-hand side of the same equation, we obtain 
Bp:m)^P 0 + m {fe 1 P I +«^ i P 1 +«BJP 1 +UBJP, } 
: m)B y B 2 P 0 +m{?t/P 1 d^B y P 1 +;yB 2 P 1 -f UB y B x P, j- 
Hence 
m _ 4 H 0 (P£ : m) 
— Bp? 0 : m+^Pj+toBsPj+ttBaPi, B 2 B y P 0 : m+wTj-Hd^Pj+wByP!, . . 
B^BJPo: m+w'Pj+wB^Pj-f flB 2 P„ B*P 0 : m+ yJ^-j-vB^Pj + vB^P,, . . 
: B 2 P 0 : m+WjPj, 
B 2 B y P 0 : m-J-w'Pj . 
• b 2 P n 
u 
B y B 2 P 0 : m+w'P, 
ByP 0 : m+^j Pj . 
• V u 
V 
B s B 2 P 0 : m+i/ Pj 
B 2 B^P 0 : m+w' P! . 
• BJ\, 
w 
B*B 2 P 0 : m -J- V P! 
■B*B y P 0 : m-fm'P, . 
. B,P„ 
Jc 
V. 
-1 
u 
V 
. -1, 
1 
But if p 0 , p y , represent the degrees of P 0 , P„ respectively, we have 
#(col. l)+y(col. 2)+3(col. 3)-J-£(col. 4)— (n— l)Pi(col. 6) 
=(Po— 1)S*P 0 : m 4-(w— 1 )mPj — (n — 1)mPj = (^ 0 — l)B a .P 0 
(#,-l)B,P 0 : m+(rc— 1 )®Pj — (n—l)'vY 1 (p 0 — l)B y P 0 
(Pi+n- l)Pi (Po+n — !) p i 
0 0 
But if P be of the same form as T in this section, or of any form having u, v, w, Jc 
as its first four columns, then at the principal points (B 2 , B y , B 2 , B f )P 0 =0; and conse- 
quently all the terms of this column will vanish except the fifth, which will 
=(jp 0 +»-l)Pi- Further, if we operate in a similar manner upon the lines, viz. if, for 
line 6, we write 
#(line l)+y(line 2)+2(line 3)+£(line 4)— (n— line 6, 
the whole expression will =(p y -\-n— l)\n — 1) _2 X 
B 2 P 0 : m+^jPu B a B y P 0 : m+w'Pi, • • 
ByB^o : m-f w'Pj, B*P 0 : m+^ P„ . . 
B a BJP 0 : m+t/ P 1? B 2 B,,P 0 : m^^' P 15 • • 
B^JA : m+? P„ B*B y P 0 : m+m'Pu . • 
