1356 
draw a line perpendicular to 6, and o,. Let 5, be the point, where 
it cuts this last. plane, the “base”, and A, the point where this plane 
is encountered by the generating line through A,. If then “ A,A,b,=9%, 
we have 
A,B, = A,A, cos 9 KAD Ey ns Se a a ee 
The strokes over the letters indicate the absolute values of the 
distances A,5, and A,A,. 
It can be shown (§ 8) that, all quantities being expressed in natural 
units, the “volume” of the prism P is found by taking the product 
of the numerical values of the base 5, and the “height” 4,5, 
Let now linear three-dimensional extensions perpendicular to A,A, 
be made to pass through A, and A,. From these extensions the 
lateral boundary of the prism cuts the parts o,' and o,' and these 
parts, together with the lateral surface, enclose a new prism P’, the 
volume of which is equal to that of P. As now the volume of P’ 
is given by the product of A,A, and o,', we have with regard to (13) 
dd cas. 
If now we remember that, if a vector perpendicular to o, is 
projected on the generating line, the ratio between the projection 
and the vector itself (viz. between their absolute values) is given 
by cos ® and that a connexion similar to that which was found 
above between a normal section 9’, of the prism and 6, also exists 
between o’, and any other oblique section, we easily find the 
following theorem: 
Let o and o be two arbitrarily chosen linear three-dimensional 
sections of the prism, N and N_ two vectors, perpendicular to 5 
and o resp. and of the same length, S and S the absolute values 
of the projections of N and N on a generating line. Then we have 
So 66200" See eee 
§ 19. After these preliminaries we can show that the left hand side 
of (10) is equal to 0, if the numbers g,, are constants and if moreover 
both the rotation R, and the rotation R, are everywhere the same. 
For the two parts of the integral the proof may be given in the 
same way, so that it suffices to consider the expression 
fire Nido DR ale 
Let X,,...X, be the components of the vector N, expressed in 
v-units. From the distributive property of the vector product it then 
follows that each of the four components of 
