PEOFESSOE CAYLEY ON CITEYATT7EE AND OETHOGONAL SXJEFACES. 243 
32. We may write 
A»=0A+2(^.,?- S v?)(Z5Y-Y8Z) + |{«A+A}, 
B"=0B+2(~ ( y- s A)(XSZ-Z5X) + A«B+B}, 
C r '=tfC+2(ii ? - ? |f)(YSX-X5Y) + |{ 4 ,C + C}, 
F"=dF+ (V--S- v?)( x5z - ax )+ (^?-^(VSX-XSY)+${»F+F}, 
G» = 0G+ (^- s ^-)(YSX-XSY) + (id: 4 -^(ZSY-Y8Z) + ^{ a G + G}, 
H"=0H+^- S w)(ZW-YSZ) + (^- ? A)(X5Z-ZSX) + ^{<»H+H}, 
iii which equations A, 13, &c. are the like functions of a, b, Sec. that A, B, &c. are of 
«, b, Sec.; viz. A=2hZ — 2gY, See. 
The value of AY is 
AY=2Z('|-|(/ ifi ,+A)+^mYAf( SY< ^ + 5X< y- V( Wi.f) 
- 2Y (| - 4( ? »+ 7j) + ^SZSX} -f(® 4f +«£f)- W*) 
—2V(hd r —gi,)g, 
which is 
= — -|0A+ A) + 1! &X(Z&Y - Y^Z) 
--~(Zd :l -Yd,) g -ZV(hd,-gi,)z 
- ~{72Y-YlZ)d^ - 2Y(Zd y -Yd z )d xi . 
Hence the value of A" is equal to the last-mentioned expression, together with the 
following terms : — 
+| (AL+A)-fi&X(ZSY-YSZ) + | (ZSY-Y&Z )d& 
which destroy certain of the foregoing ones ; viz. we have 
A"=(2V ff -^)^-2(v/ l -A-)^-2V(Zrf,-Y ( yig. 
