392 PROFESSOR A. R. FORSYTH ON THE DIFFERENTIAL INVARIANTS 
We then modify this set of four, and replace H (^¥" 3 ) and (^¥" 3 ) by and ( 5 . 
where 
= (Em' - 2 F/' + GF) + ... 
@2 = (E (En' - 2 Fm' + G^') - F (Em - 2 F/ + Gk)] xPoi + ... ; 
and the set W" 3 , J (Wo, W" 3 ), (J, replace <, H {w",), cp (u^'g), J (ic,, lo",) in the 
aggregate, which remains complete after the change. The set of equations, which 
exhibit the equivalence of the four inserted forms to the four ejected forms, is 
simdar to the corresponding set in § 42 ; it is more complicated because the ground- 
forms w^", W "3 are of tlie third order. 
The geometric significance of the four inserted forms can be obtained from the 
consideration that they stand related to the curve ^ = 0 exactly as ic'g, J (?Co, 1 F 3 ) 
Cj, Co to the curve (j) = 0. Adopting the notation of § 51, we thus have 
All otlier properties of the curve 1 // = 0 up to the order retained can be expressed iu 
terms ot the invariants of the aggregate; the examples given in § 51 will be a 
sufficient illustration of the remark. 
T/u 
Aggregate for a Single Curve (f) = 0 up to the Order Retained. 
53. Ihe 29 invariants in the preceding set have a closer affinity to the curve ^ = 0 
than to the curve A = 0, the chief reason being that the first derivatives of (/> were 
made the variables for the binary forms. By taking the first derivatives of i/; for 
these variables an equivalent set of 29 invariants could be obtained, having a closer 
affinity ^ to the curve i// = 0 than to the curve ^ = 0. And it would b^ possible 
to modify each of these two sets, so as to construct a new equivalent set of 29, 
symmetrically related to the two curves. All that is necessary in each modification 
is to secure that the retained aggregate remains algebraically complete. 
Out of tlie set of 29 invariants retained, there are 20 which are not affected 
by the curve i/i = 0 in their expression; and therefore we infer that all the 
differential invariants of a surface and a curve <^ = 0 upon the surface, involving 
derivatives of (f) up to the third order inclusive, involving the magnitudes E, F, G 
and their derivatives up to the second order inclusive, involving also the fundamental 
