ME. HOPKINS ON THE THEOEY OP THE MOTION OF OLACIEES. 
697 
K, [Jij, V being the angles which its direction makes with our present coordinate axes of 
X, y, z respectively. Hence 
=A?cos^a,-l-BiCos^/3i+C, cos^yi, 
Aj cos aj 
COS X ^ j B2 cos^ /3j + Cj cos^ 7j ’ 
Bj cos /3j 
cos fJj ^^2 gQgS J 52 gQg 2 _|_ Q 2 gQg 2 
C, COS y, 
COS V 2^ — 
V Aj cos^ + Bj cos^ /Sj + Cj cos^ y j 
But j3i = 90° in the case before us, and therefore yj must =90°, ^. e. the direction of 
the whole resultant force {p) on the small plane s must be perpendicular to the axis of 
y, and must lie in the plane of xz. Thus, the 
plane of the paper representing that of xz (fig. 1), 
the direction Op in that plane may represent the 
direction of p ; and if Om bisect the angle be- 
tween the coordinate axes x and z, that line will 
be the trace of a plane on xz coinciding with the 
plane of s, and therefore perpendicular to that of 
xz. Consequently, if we resolve the whole force 
p normally and tangentially with reference to the 
plane s, the tangential part will evidently coin- 
cide with Om. But, from the particular values 
of j3,, and this tangential force must neces- 
sarily be the maximum tangential force Tg. Con- 
sequently, if we call the axis of y the axis of mean principal tension or pressure, the 
direction of the maximum tangential force (Tg) will be perpendicular to this mean axis, 
and will bisect the angle between the other two axes of principal tension. 
19. Hence, in our first problem, the equations (2.) and (c.) determine a, j3, y, and 
The cubic for finding j?, which is deduced from them, shows that there are three values 
of that quantity, the three principal tensions, whose directions are defined by corre- 
sponding values of a, j3, and y, and which are at right angles to each other. These 
quantities being known, the value of Tg, the maximum tangential force at the proposed 
point, and the direction in which it acts, are immediately determinable from the results 
of the second part of the problem as given above. To do this we must first determine 
the three systems of values of a, /3, and y which have been denoted by osi, j6i, and yi, and 
which determine the positions of the axes of principal tension ; and also the values of 
the three principal tensions which have been above denoted by A,, B,, and Cj. For the 
greater simplicity we then take these axes of principal tension for those of x, y, and z. 
If Ai, Bi, and Cj be in order of algebraical magnitude, the axis of y will be the mean 
axis. If Ai be a pressure and therefore negative^ the proper order will be B,, Cj, — Aj, 
Fig. 1. 
