IME. W. SPOTTISWOODE ON THE CONTACT OE CTJEVES. 
51 
Also writing 
we have 
also 
^ BU , ^ BU rr 
dy + dz — 
-n= 
=an. 
1 
□ 
■xnv 
=/3n. 
1 
□ 
B^U 
B"U 
BU 
dx 
dxdy 
dxdz 
dx 
cc 
B^U 
B"U 
BU 
i3 
dydx 
dy^ 
dydz 
dy 
B"U 
B'U 
BU 
dzdx 
dzdy 
dz^^ 
dz 
7 
BU 
BU 
BU 
0 
0 
dx 
dz 
a 
7 
0 
0 
TT 
where H is the Hessian of U. The corresponding reduced form of this expression, 
i. e. when a=l, (3=0, y=0, is 
/duyd^u o Bu 5u yu. /Buyyu _^tt 
y y dy^ dz dy dydz'ydy j dz^ (n— 1)^ 
And by means of these we may deduce the following : — 
ioDv—vTiw={n—V)~\Tr'B., wD{o — l)“^ya’H, wD^v —v'D^io={n—\)~‘^zx3i, 
u'Dio—vjJ)u={n—\)~^xyll, 2 tDiW— 2 (;Dp^=(?i— H, wDgW— ^yD 2 ^^ =(n— 
vJ)u—u'Dv={n—l)~^xz'ii, ^;DiW —u'DdJ={n—l)~^yz31, vJO^u—ii'D^v =(w— 1}~VH. 
Again, 
2/D2H-zD,H=3(?i- 2)^H, 2DH-.YD2H=3(n-2)^H, a;D,H-3^DH=3(w~2)^] 
7jJ)^u—z'Ddi={n—3.) , zDw—xD^io=x'D,v—y'Dv={n—l)^ 
To which may be added, 
□ XD u-Y DyJ) v-\-nvJ0 w=lB.\, 
□ XDiZi+ □f'D,w=SH|W;, 
□ ADjiid- □iM'HjVd- OvD 2W=lIlv, 
itDv-^vDiV-\-wJ)2V=0, 
Wu = — (n— 1)"^^ (^H) — 3 Aw + 3.rH, 
jy-v =-(w-l)-^|(^^H)-3A^, 
= „ (w - 1)-=^ - 3 Aw, 
n 2 
