CONTACT AT ANT POINT OF A PLANE CUEYE. 
383 
where H is a homogeneous function of the degree 3m — G; since the x is not 
affected by the differentiation, must be treated as of the degree 3m— 7, and 
for tlie like reason, stands for ; we have 
(^b,+2bJH=(3m-6)(3m-7)H 
— 2( 3m — 7 
In like manner, 
= (3m-7)(7^b,+(gb,)H-Ab.,(lb,+(ab,)H; 
and hence 
(m-l)^(Cb,-Bb.)^H 
= -^=(35, jr, CI3„ SJ’H 
+2.T[(3m-7)(S3,+©3.)H-x3,(®3,+®3,)H] 
_3[(3m-6)(3m-7)H-2(3m-7)^3,H+.t”33-I] 
= -x%% 2, C, 4f, 0, ®I3„ 3„ 3,rH 
+ 2(3m-7)x(a3,+®3,+@3.)H 
_(3m-6X3m-7)gH. 
25. The other terms are formed in a similar manner; and collecting all the terms, we 
liave 
(m-l)^D^H=-(3m-6X3m-7}(a B, C, #, 0, IB, 
+ 2( 3m — + vz){{^X + JT + + ((§A + + C 3) 
3S, C, f, (3, lib., b„ b JH, 
or, attending to the signification of the symbols b-, ff>, □ , O, this is 
(m-lXD'H=-(3m-6X3m-7)HO 
+(3m— 7)^n 
Proof of the expressions for (m— IXR^, (m— 1)^E4: — 
26. 5Ve have 
(m-lXRl=(m-l)XDPIX 
= -9(m-2yH^0 
+ 3(m-2)Hn^ 
(m-lXE4=(m-lXD^H+(m-l)an-(3m-6)(m-l)™ 
= _12(m-2XHa> 
+ 4(?/^— 2)Dfy 
wdiich are the expressions required. 
3 E 2 
