PROFESSOR CATLEY ON CURYATURE AND 0RT1I0G0YAL SURFACES. 247 
we find without difficulty that the terms in question (being, in fact, the complete value 
of the expression) are 
=v((A), . . .1 a,, d z )\ 
39. As regards the terms involving the first differential coefficients, observe that the 
whole coefficient of is 
- 2(5)(V/-?A) 
+2W(V^) 
+2(/)(v(i-0-4(YW-ZSZ)) 
+2 & )(VA-^) 
-2(A) (V?-^), 
which is 
=2 V {(# + (/)} +(0 ? - ((%+ (A)/-K/)0 } 
+ Z((A)SX + (708Y+(/)iSZ) - Y(0)SX+(/)3Y + (cjiZ) } . 
40. The reduction depends on the following auxiliary formulae : — 
(o)+A(A)+^)=vSy-xSx, 
„ +A „ +/ „ = — Y§X, 
?3 ~rf n +<? » = — Z&X, 
a{h)-\-h(Ji)+g(f) = — X<SY, 
h „ + A „ +/„ =VdV— Y«$Y, 
( j ii +/* 3) + o „ = — Z§Y, 
«(<?)+' Hf)+ff(c)= -xiz, 
A ,3 fi-A „ -\-f „ = — Y§Z, 
9 33 +/ 33 +C„ = VH — ZoZ, 
where, for shortness, I have written <$X, AY, AZ to stand for «X + AY -)- t/Z, AX + AY ffi/Z, 
yX+/Y+cZ respectively, and YAV for XAX+YAY+ZAZ, (=«, . . YX, Y, Z) 2 . 
From these we immediately have 
(«)AX + (A)AY + (t/)AZ = Y(XA V - VAX), 
(A)AXH-(A)AY+(/)AZ=V(YAV-VAY), 
(«)AX -f (/) AY -j- ( c) AZ= V(ZAV - VAZ). 
Hence, in the coefficient of d x §, the first line is 
=2V(-YAZ+ZAY), 
and the second line is 
=Y{VZ(YAV-VAY)-VY(Z~AV-VAZ)}, =2V(YAZ-ZAY) ; 
so that the sum, or whole coefficient of is =0. Similarly, the coefficients of d,o and 
d,o are each =0. 
9 t 9 
j li J 
