336 
PROFESSOE A. E. FORSYTH ON THE DIFFERENTIA!. INYARL4NTS 
holds for all values of dx aud dy ; so tliat the laws of transformation for L, M, N 
ai'e the same as for E, F, G. Hence 
r/L 
d! 
r/M 
lU 
If) I 
2i\r77^„ 
-+ M-) 7 oi + ^Vio r 
c/N 
df 
= 2Me 
01 
+ 2 N 77 
01 
Usinu the relation 
(P, Q, K, SXdx, dyf = d.^ 
d /F' 
in the same way, we find 
__ (/P 
dt 
dO. 
ds \p 
3Pfif. 
= (P', Q', P/, S'XdX, dYf 
d-3Qr? 
10 
— ^7^’ — + 2Q^]n + Q’?oi +2 Pi.77j^q 
dPv, 
df 
dS 
df 
— 2QX11 + 2Pi77m + Sy 
10 
voi 
+ 381701 
Plsinu' the relation 
(a. 13, y, S, ejdx. d,/)* = . A T) = (,', y, </Y)' 
similarly, we find 
da 
(If 
+ 4a^]0 + d/3r7|Q 
= «Gii + 3 /S^oi + ^’7(11 + 3 ). 77 
+ 2yf,n + 2717,11 + 25 
= 57&1 + 5f,„ + 3S77 „i + €17 
10 
^10 
10 
(h 
df 
= ^K 
01 
+ ■le77 
01 
And so for the increments of the other fundamental magnitudes. 
8. The increments of the derivatives of E, F, G are required; they can he 
obtained liy the following method, differing from that which is adopted by Professor 
ZoRAWSKi. Let X and y become x + h and 3/ + A’ respectively, and let the con¬ 
sequent new values of X and Y be X + H, Y + K; then 
