AND DETEEMINANTS OE LINES. 
481 
We have therefore the equation 
D,{det(P, Q)}=det {P, D,(Q)}+det |D,(P), Q}. 
The same equation may also be proved by considering the algebraical determinants 
which represent the projections of det (P, Q) ; and it may in fact be easily deduced from 
the following identical equation, 
29. It may be here observed that the formulae (I.) and (II.) in section 26, and the 
formulae (I.) and (II.) in section 28, show that there exists an intimate symbolical connexion 
between det (P, Q) and the product P, Q. In fact the only difference between their 
symbolical properties consists in P and Q not being commutative in the expression 
det (P, Q), and being so in the expression for the product. 
30. There is one more proposition which is often very useful in analytical dynamics. 
Let it be required to find det (R, Q'), where Q' itself equals det (P, Q). Let the required 
line det (R, Q') be denoted by U. Then, by the definition of a determinant, U is perpen- 
dicular on R and on Q', which last line is itself perpendicular on the plane containing 
P and Q. Hence it follows that U is perpendicular on R and in the plane containing 
P and Q. 
We have still to find the magnitude of U. For this purpose let the angle which R 
makes with the plane containing P and Q be \j/, so that 4^ is the complement of the 
angle between R and Q'. 
Moreover, let & be the angle between P and Q, and let the magnitudes of P, Q, Q', R be 
denoted by y, q’, r respectively. Then, since U is det (R, Q'), it follows from the defini- 
tion of a determinant, that the length of U equals r^'sin or rq' cos x}/. Simi- 
larly, q'—pq sin 4 Hence the length of U equals pqr sin 6 cos \f/. 
There are two cases especially which frequently occur in dynamics, first, when R is 
identical with Q, and secondly, when R is perpendicular on Q. 
Let us first take the case of R being identical with Q ; then and r~q. There- 
fore the lequired determinant is a line in the plane containing P and Q, and perpendi- 
cular on R or Q, and its length equals pq^ sin 6. 
If, moreover, Q is perpendicular on P, then the required determinant is in the 
direction of P, and its length equals p^, since that, if P is perpendicular on 
Q, we see that det -jQ, det (P, Q)|- is a line in the direction of P ; and therefore evi- 
dently det |Q, det (Q, P)} is a line pq^ opposite to P. 
Let us now take the case of R being perpendicular on Q. Then it might be easily 
proved by spherical trigonometry, that sin & cos 4/ equals the cosine of the angle between 
R and P. But we will prove this by analysis, because in doing so we shall meet with 
formulae which will be of use in the sequel. 
