724 
MR. A. McAULAY ON THE MATHEMATICAL 
52. In (2) it is to be noticed that one more variable, viz., q, occurs in l' than in 
l or l". The reason is obvious, but on account of the fact, it is easiest to arrive at 
formulas transforming differentiations of l into the corresponding ones of V by first 
considering the similar relations between l and l". 
Let a and r be taken as a typical independent variable intensity and flux respec¬ 
tively. V is obtained from l" merely by changing every c t" and r" into q~ l cr'q and 
q~ l r'q respectively. (§ 7.) 
By considering the increment in V and l" due to an increment in a a" or t" , we at 
once obtain 
yi' = q yr q ~\ yr = q yrq- 1 .(4). 
By a similar process it is easy to see that 
3/ j _ dl' , , 
~ d s = gj 9 ClS 
yds = yds 
VI = V (ini') 
(I V 
Dr 
= m * 
= jVW 
= V' (ml") 
= ar 
(5). 
53. We proceed to find the corresponding relations for the other variables. Let us 
in l and l" vary T and every cr and r, and cr" and t". Thus. 
— SSSo-^VZ — ]£SSt t W — SST'^GZZ; — S l — l '8m -{- mSl'' 
= l"Sm + m {- SSSo-'VVT' - %SSt" 7 V'1" - SS¥£(ir£}. 
Now (§7) 
Hence 
Sr" = m~ } (Si/s — m~ l Sm.xjj) t -f- m~ l xpSr. 
Scr' — — xjj~ l 8xfj\p~ i cr -f- \jj~ ] Scr. 
Substituting these values and equating the vector coefficients of the arbitrary 
vectors So- and Sr, we obtain 
y = y'l" = m x - L yi' 
y = xjy'i" = x 'yr . . 
• ( 6 ). 
• (7), 
the last result in each of these being given by equation (4). These equations show 
that y, „V7', y'l" bear to one another exactly the same relations as r, r, r", which 
