280 
PROFESSOR A. R. FORSYTH ON THE 
derivatives of such a function as 0 vill occur, and tlie order of the derivatives vdll 
depend upon the order of the derivatives of a, h, c, f, g, h which arise in the invariant. 
Clearly one invariant is provided by the determinant 
r. 
0 
a, h, g ; 
A, h, f 
for on substituting the values of tlie fundamental magnitudes in terms of the 
derivatives of x, y, z, we have 
d{u,v,w) 
= ni/, 
so that L is an invariant for v hich the index /x is 1. 
It is convenient to denote the minors of the constituents in by A, B, C, F, G, H,, 
where 
A = bc — f~, B = ca — g"% C = ah — Jr. 
F = gJi — af, G = A/ — hg, H =fg — ch. 
4. The laws of transformation of the various magnitudes are required in order to- 
construct their increments which correspond to the increments of v, r, w. AVriting' 
f/s- = (o, h, c, j\ g, dv, dwY 
= {a , h', c', g\ li'ydu\ dv’, div')~, 
-dll' dv' div'\~ 
we have 
a 
/ / 7 / , did'' 
= «'(! + 2^,00 “I" 2A'>7ioo<^^ + 
— a! A- (2«|^ioo + 2/i>7ioo + 2p'^ioo) 
on neglecting squares and higher powers of dt. Similarly 
^ + (2^^^oio + 2^01^ + 2/4io) dt, 
c = d -\- (2p^()|^ + 2 /77001 + 2c{^ioi) rZf, 
/ = /^ + {^i^ooi + 9^\m + ^^^ooi +/(’?oio + ^oui) + 
9 = 9 ' {«^ooi + ^^^001 + 9 (fioo + ^001 ) +/? 100 + ^^loo) 
Ji = h' -f [a^oio + (^100 + "^oio) + 5''Ciio + ^^100 +/^ioo} 
These six equations at once give the increments of a, h, c,f, g, Ji, They can also be 
used to construct the increments of the derivatives of fundamental mao-nitiides, as 
follows. 
