OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
341 
— + <f>uiVio 
— = ^loioi + 4>inVoi 
dt 
— 2^00^10 + ^in6.o + 2 ^ii17io + 4^Q\'n-y^ 
— = <^11^10 + <^20^01 + 4>in^n + </>o2’?io + ^ii^oi + 4>()\V\\ I' ^ 
^^4*02 _ 
dt 
dl 
#21 
dt 
dt 
d<pQ.^ 
dt 
— 2'^ll^01 + ^10^02 + 2(^02^01 + <^01^02 
= ^4^3o4iq + 3^20^20 + (f^in^so + 3</>ci’?io + '^^4^nV2o + '/'oi^so 
= ^4^2i4lO + ^.30^01 H“ ^Il4o + 2 <i!>20^ii 4- <^10^21 
+ 2(^i.3'>7io + 4^2\Vo\ + 4>o2V2() + '^'f^nVn + (ko\V2i 
= <^12^10 + '^2\^(n + 2<^n^ii + (f>2(4a2 + 4^v4v2 
+ ^kzVV) + 2</>13^01 + 2<^02^11 + 4^uV02 H- 4>o\Vu 
= 3 <^ 12^01 + Hu4o2 + 4^w4o3 + 3 (^ 03^01 + 3 </> 02^02 + </> 01^03 
12. A comparison of the expressions of the increments of the derivatives of E, F, C4 
on the one hand, and those of the derivatives of a typical function (f) on tlie other, 
leads to one immediate inference as to the arguments that enter into the composition 
of a differential invariant. Suppose that such an Invariaiit is rerpiired to involve 
derivatives of a function (j) up to order M in x and y combined ; the increments of 
these derivatives involve (among others) the fpiantlties 
^MO) ^-MU • • • 1 ^O.M ; ym ; • • • > 
The invariantive property requires that the terms involving these quantities must (if 
they do not balance one another) he balanced by other terms involving these same 
quantities ; and therefore derivatives of E, F, G uj) to order M — 1 in x and y 
combined must occur. And conversely. 
In particular, if derivatives of (f> of the third order occur in an invariantive 
function, it must contain derivatives of E, F, G of the second order. 
The Ditferentiul Equations Defirtiiuj the Invariants. 
13 . The invariantive property is used, exactly as in Professor Zoeawski’s applica¬ 
tion of Lie’s method, to obtain partial differential equations of the first order 
satisfied bv any invariantive function. We proceed from an equation such as 
f = -^"7; 
