332 
PKOFESSOK A. E. FOESYTII ON THE DIFFERENTIAL INYxVEIANTS 
Ihese six quantities can occur in the invaiiantive function required, as 'svell as their 
derivatives of any order with res 2 )ect to x and y. 
But there is a difficulty as regards the derivatives of L, M, N ; for tliere are two 
relations, commonly known as the Mainaedi-Codazzi equations, which express 
aL _ aM^ m _ aN 
dy dx dy dx 
in terms of L, M, N, E, F, G, and the first derivatives of E, F, G. To avoid this 
difficulty, it is convenient to introduce the four fundamental magnitudes of the third 
01 del, denoted by P, Q, B, S ; the six first derivatives of L, M, N can he expressed 
in temis of P, Q, B, S linearly, together with additive combinations of L, M, N and 
of the first derivatives of E, F, G. 
The second derivatives of L, M, N will thus he expressible in terms of the first 
deiivatives of P, Q, B, S, together with the ajipropriate additive comhinations Ifee 
from those derivatives. But again there is a difficulty as regards these; for there 
are three relations, which exjiress 
aQ_ap^ as _ aB 
dx dy dx dy ’ dx dy 
in terms of P, Q, B, S, L, M, N, E, F, G, and the first derivatives of E, F, G. To 
avoid this new difficulty, it is convenient to introduce the five fundamental 
magnitudes of the fourth order, denoted hy a, /B, y, S, e; the first derivatives of 
P, Q, Pt, S (and therefore the second derivatives of L, M, N) can be expressed 
linearly in terms of a, y, S, e, together with additive combinations of P, Q, B, S, 
L, M, N, E, F, G, and the first derivatives of E, F, G. 
And so on, for the derivatives ot successive orders of L, M, N ; we avoid the 
difficulty of linear relations among them by the introduction of the successive 
fundamental magnitudes. The analytical definition* of these magnitudes can he 
taken in the form 
where p is the I'adiiis of curvature of the normal section of the surface throiiP'h the 
See a paper by the author, ‘Messenger of Mathematics,’ vol. 32 (1903), pp. 68 d acq.; see also 
§31, pod. 
