PROFESSOR A. R. FORSYTH OX THE 
wliere, after .some rerliictiou, fp i.s found to be mven 1 
)y 
<1> 
cr 
1 _ .) :■! ^"bc I 0 _i_ o \ 
A, 
A, 
A, 
AToreovei' 
_ -v ^E Ai. — AA-)^ 0 )oA2i — A.^be 
+ 
A 
-^1 
a — O', h , g- 
h , b -- 0-, f 
f , c — cr 
_ Ao Aoi I Aio o o 
- a7 “ " + A = ~ ■ 
cr 
to 5 
conse<[ueiitly 
1 , A., A.1| . A].t 0 
= CT* — 2cr® + .y (A|j@i. - + 3A,0j) 
- 1 ] 
2cr 
1 
5(^'b"^12 “ -^2®l) + T G 
iAi 
AA^hen cr = o-,, we may take 
Avhen a = cr^, we may take 
^ d(f) d(f) ds^\xln) ’ 
dn dn 
p = p, = = A. (#') • 
‘ d<i> d^dsjdnj' 
dn dll 
I he two equations which thirs arise b}’' taking cr — cr^, and cr = ct.^, when combined 
symmetrically aiid rationally, may be regarded as two equations defining Aa^ and A., 
algebraically in terms of the quantities and 4- 
«s, \dn/ ds .2 \dnj 
34. Tlie geometric sigjiificance of most, if not all, the difierential invariants of the 
second order, which occur (§21) in the algebraic aggregate Avhen two surfaces are 
given, can be obtained by noting the properties for the second surface corresponding 
to those discussed for the first surface ; they will not be considered further in this 
memoir. 
Nor is it my intention in this place to discuss the geometric significance of the 
sixteen members ot tlie algebraic aggregate of difierential invariants up to the third 
order inclusive which occur (§ 28) when there is ouly a single surface. But oue 
