CONTACT AT ANT POINT OF A PLANE CUEVE. 
37 
•5. Next, preparing to substitute in the equation 
D^U-n.DU=0, 
the consecutive value of DU is 
(a “h dx ^d^x -j- ^d^x -f- -}- &c. 
= (^0 + ^lH“i^2+6^3+ 2\^4)U, 
where 
BoU = ( aB^+ 3/B2,+2B^)U =mU. 
Eeducing by the above results, the consecutive value of DU is 
= _iBfU-i(B?+3B.B,)U-^(Bt+6B?B,+4B,B3+3B^)U. 
6. Hence also writing 
P=«A +<7^, 
'dj^=adx ■\-hdy -\-cdz, 
B 2 P = ad?x -\-hdHj-\- cd^z, 
the consecutive value of — HDU is — (P+BiP+-|-B 2 P) multiplied into the consecutiv 
value of DU, and the product is 
=P. iB^U 
+P.i(Bf+3BA)U +B,P.iB?U 
+P.^(BH-6B^B2+4B.B3+3B^)U+B.P.i(B?+3BA)U+iB2P.iS?U. 
T. The consecutive value of D^U is 
= {x-\-dx-\- \d^x + + -^d^xfbJ^V + &c. 
= a" 1 
which is 
and obserdng that 
+ 2xdx 
+ a(Z^a4- {dxf 
+ ^ A<Z^A+ dxd^x 
■ B"U+&c., 
+ -^xd^x -^^dxd^x + ^{d'^xf 
= 
+ 2BoB, 
+ B0B2+B? 
+ ■3^0^3+^1^2 
4-lV^0^4 + i^l^3+4^2 
U; 
Bo U=m(w— 1) U, 
BoB,U= (m-l)B,U, 
BoB 2U= (m-l)B2U, 
BoB 3U= (m— IjBjU, 
BoB,U= (m-l)B,U, 
3 D 
MDCCCLIX. 
