334 
PilOFESSOR A. E. FOESYTH ON THE 
DIFFEEENTEAL INVAELINTS 
Hence we retain none of the third class of 230ssible magnitudes. But after the 
reasons adduced, we should only be justified in dropping y' from the set of magni¬ 
tudes when it was otherwise required, if we associated the first derivatives of the 
appropriate function i// with the functions already retained; or in dropping y'\ if ve 
associated the second derivatives of with the functions ah’eady retained; and so 
for the other derivatives of y. (An example occurs later in § 24.) 
Note.—I n calculations subsidiary to the determination of the geometric 
significance, it is found necessary to use the relations involving the derivatives 
ot L, M, N, P, Q, Pv, S ; it may therefore be convenient to give their explicit 
expressions.^' They are :— 
where 
P = L,o - 
2 (Lr + Ma) 
II 
tp 
o 
1 
2 (Lr -f Ma') 
• 
1 
o 
II 
(LP -h Ma') 
- (Mr + Na) 
E = Moi - 
(Lr" H- Ma") 
- (MP -1- Na') 
- 2 (MP -f- Na') 
S =Noi 
- 2 (Mr" + Na") 
2VH = GEj,-F(2F,,-Eo,)1 
2W =:GE,,-FG,, I, 
2V2P' = G(2Foi-Gio)-FGoJ 
2V^A = E (2F,o - Eoi) - FE,o ■] 
2V2a' EG,o - FEo, 
2VW^ = EG,,-F(2Fo,-GJ^ 
a = Pj, - 3 (Pr-f qa) 
/3 = P„J _ 3 (PP + Qa') - f (FL - EM) 
= Qio- (Pr'-fQA') -2(Qr-fPA) +^^ 2 (FL-EM) 
y = Qoi - (Pr" -f Qa") - 2 (Qr' + Pa') - i (GL - EN) 
= Kio - 2 (Qr' + PvA') - (Er -f Sa) +1 (GL - EN) 
S = - 2 (Qr" + Ea") - (EP + Sa') - i (GM - FN) 
= Sjo - 3 (EP -f Sa') + I (gm - FN) 
e = Soi - 3 (Er" -f Sa") 
where = LN — M^. 
They are quoted from the author’s paper, mentioned in ^ -1. 
