ME. HOPKINS ON THE THEOEY OF THE MOTION OF GLACIEES. 
723 
flank on the left, the plane of xz. The plane of yz will be a vertical transverse plane ; 
let it be represented by the plane of the paper, to which, therefore, the line of motion 
of every particle will, by hypothesis, be perpendicular. Let P (flg. 14) represent a 
Fig. 14. 
A y TA 
material point in the plane of the paper, or that of y z. (We take it in this particular 
plane because we shall not be immediately concerned with the coordinate x.) Let 
AM=?/, and MP= 2 :. Also, take another point p supposed to be very near P, and let 
its coordinates be y-\-ri and z-\-^, y; and ^ being referred to P as origin. Also let V be 
the velocity of P perpendicular to the plane of the paper, and V+?; that of ^ ; V will 
be some function of y and z ; and v a function of yi and As we have chosen the 
coordinate planes, V will increase with y and z, because the centre of the mass moves 
faster than its sides, and the upper moves faster than the lower surface ; and for the 
like reasons, v will increase with and Now since jj and ^ are very small, we may 
consider any increase of v, due to an increase of as and an increase due to that 
of as where jM/ and /«/' are functions of y and z, but independent of and 
They express the rate at which V increases as we pass from P to contiguous points in 
the plane of the paper, and in directions parallel respectively to the axes of y and z. We 
shall then have 
or if 
ri=^cos 0, 
sin 6, 
v=§(ij!j cos sin 0). 
From what has been above stated, this quantity must be a maximum with respect to 0, 
considering fji,, yJ, and as constants. This gives 
— yj sin B-\-yJ cos S=0. 
♦ The tendency of the tangential differential motion of two particles very near together, to bruise or dis- 
locate the mass, will manifestly depend on the contortion or twisting produced by this relative motion, and 
therefore on the angular change of position of the line joining the two points. Thus, in fact, f is the quan- 
tity to be really made a maximum. This is equivalent to considering § constant in differentiating as in the 
text. 
