690 
MR, A. McAULAY ON THE MATHEMATICAL 
I have not hesitated to put the symbolic vector, V, among the intensities since it 
obeys all the laws thereof. The definitions of the connection between cr. cr', and c r", 
and between r, Y , and Y, may be taken as the first and third of equations (5) and (6) 
respectively. The second and fourth equation of each set are easily deduced from 
these by observing that dp' = yc/p, ^ — m X~ 1 ^ [equations (26) and (37) of former 
paper], and that both dp and dt, are arbitrary vectors. 
It should, perhaps, be noticed that these connections between a,a, and cr" and between 
r, Y, and t", although very useful and intimately connected with the physical nature 
of the vectors indicated are, after all, only definitions, and thus the phrase “ where 
such and such a symbol is defined as a flux ” will frequently occur below. This merely 
means that, having assigned the meaning of one of the three vectors, say t', by 
a physical definition, the allied symbols, r and t", are defined by saying that the 
symbol in question is a flux. 
The connection between cr and cr' may be put in words, thus :—If cr be an intensity , 
any line integral of cr' referred to the present position of matter is equal to the corre¬ 
sponding line integral of cr referred to the standard position of matter. Of course, by 
the word “corresponding” it is implied that the two line integrals are to be taken 
through the same chains of matter. Similarly as to r :—If t be a flux, any surface 
integral of t referred to the present position of matter is equal to the corresponding 
surface integral of t referred to the standard position of matter. 
8. It is convenient to give here the following four simple but useful propositions. 
Prop. I. If a u , <Th be two intensities, X a n a/j is a flux .—By this is meant that 
Vo- s V/ bears the same relation to Xa u ab as does t' to r in equations (6). To prove 
S dScr a crb = mr^fidt'ficrafio-b [eq. (5) § 7] 
= S dfa-fcTb [Tait’s ‘ Quaternions,’ 3rd ed., § 158, eq. (3)]. 
Prop. II. If cr, t be an intensity and flux respectively, we have Sards = Sard! 
= Sa'I'cls '.—For by equations (5) and (6) § 7, ScrV = to -1 Sot, and ScYY = ScrY, 
As particular cases we have 
SBHcZs = SB'H'cZs', SCAcZs = SC'AkZY, SD0c/s — SD'0'cZs' . . . (7). 
Prop. III. If a be an intensity YVcr is a flux .—By this is meant that VVV bears 
the same relation to YVcr as does Y to r in equations (6). For any surface 
j"jScZ2Vcr = j'sdpcr [eq. (3) § 5 above] = jScZp'Y [eq. (5)] = jJScZS'V'Y [eq. (3)]. 
Hence, SdtXa = Sd%'X'a', or YVcr is a flux. 
As particular cases, note that if, as we shall do directly, we assert that 
4770' = YV H', B = VVA , 
