G88 
MR. A. McAULAY ON THE MATHEMATICAL 
regard to p. A, a particular form of V, is used when we wish to imply that the differ¬ 
entiations of the V are to refer to all the factors of a term. Thus 
V DAE = t 9 (VDiE)/Sas. 
If cr he an independent variable vector, a V, a A, have the same meanings with regard 
to cr as V, A, with regard to p. 
Q is a symbol of differentiation which is thus defined if trr be an independent 
variable self-conjugate linear vector function of a vector, given in terms of the scalars 
P and c by means of the equations 
mi = P i ~b N j -j- ML 
7ttj — Nt -f- Q? -j- L Jc 
ml' = Mt -f Lj -j- P/c, 
jd is a symbolic self-conjugate linear vector function of a vector given by 
2 a Qi = 2i wj + j + Jc 
0P 
_0_ 
0N 
0M 
— i~-\- 2j -Jr + k 
2 m QZ; — i 0 + y 57 + 2 Jc 
0N 
0 
0Q 
0_ 
0L 
0M 
0_ 
0L 
0_ 
0Pt ‘ 
Numerical suffixes are used exclusively to denote to what symbols the differentiations 
of a V or Q refer, the operator and the operand having for this purpose the same 
suffix. 
Let Q (a, /3) be any function of two independent vectors a, /3, which is linear in 
each Then £ is defined by the equation 
Q (£, £) = Q (Vi, pi) = Q (i, i) + Q (j>j) + Q (k, k). 
Similarly if P (a, {3, y, S) be linear in each of its constituents 
P (£n £]> £%> £ 2 ) — P (^n Pu V 2 , p. 2 ), 
and so to any number of pairs of £’s. 
At a given instant p is a function of p only, and, therefore, 
dp — — S dpV . p = y dp , 
where x is a linear vector function which is called the strain function, q, ip, ' V P, to are 
all functions of y given by the equations 
Xw — qxpivq* 1 , 
where q is a quaternion and xb a self-conjugate linear vector function of a vector. 
X being the conjugate of x> 
XX = :: ^ = % 
