240 PEOFESSOE CAYLEY ON CUEVATUEE AND OETHOGONAL SUEFACES. 
27. Taking q, % for the coordinates, referred to P as origin, of a point on the given 
surface near to P, and r/, for the coordinates, referred to P' as origin, of the corre- 
sponding point on the vicinal surface, the relation between ?/, £' and |, rj, £ is the 
same as that between dx', drj\ dz' and dx, dy, dz ; viz. we have 
§ = b(l — d x gct) — d y %n . vj — d 2 %u.%, 
?i= — d x qfi + (1 — d^(3)d — dgfi. £', 
K=—d^y • I' — dy%y. r] + (1 — (L§y)£'; 
or, conversely, 
t = (1 + d x Q a) d^a . sj - -f d z oa . 
‘1 = d x gfi •? + (! + d y %fi)r) + d.^fi . £, 
d,%7^+ d tJ oy . 7i + (1 + d^y)^ 
say ?/, £'=£+Af;, + £+A£; hence 
Xf + YV + Z'? = (X - V^)(g + Ag) + &c. 
=Xg+Y*+Zf 
+XAg+YA*+ZA£ 
— ^ + + ^f)» 
where second line is 
(Xa -f- Y/3 + Zy)(|^g -\-yid y % + d,%) 
+ § ] (X~d x a + Y dj3 + 7jd x y)<i + (XfZ (/ « + Y d y fi -f- Zt?„y)/j + (X<7.« -f- Y d.fi -f- ZYy)7 1- . 
But 
X4«+Y<?j3+Z4r= y-^3 sx=0, 
ILdyU -f- Yd y j3 -f- Zf/ y y = 0, 
Xd^+Ydfi+Zd# =0, 
or second line is=V(gc? a .g-f-J 7 ^ y g + £<2 z g) ; and we have therefore 
X'r + YV + Z'£' = Xg + Y n + Z£. 
We require 
(A', B', C', F', G', H'X^, d, ; viz., to the first order in this is 
= (A', • • 0& ?, £) 2 
+ 2(A, . . JAg, A?, A?X5, £). 
28. Here second line is 
2{(A5 + Hfl + G?)Ag+(H|+B9+E?)A,+(Gg+F9+C^)A^}: 
