724 
ME. HOPKINS ON THE THEOET OP THE MOTION OF GLACIEES. 
I^et and 6^ be the two values of 6 which satisfy this equation ; then 
and 
tan 6,= tan L— — ; 
^2= tan-' ^+180°. 
From the above expression for v, gives v a positive value, and ^ the same numerical 
value with an opposite sign, the one being the algebraical maximum value, the other 
the minimum one. 
Moreover, we see from the above value of v, that w=0 when 
tan 6-=. — = 
cot ^ 1 , 
which shows that the angular distance between the directions in which v is respectively 
a maximum and zero, is equal to a right angle. 
56. To interpret these formula, draw (fig. 15) Fig. 16. 
making the angle tan"' — with the axis of y. The 
relative velocity for p and P will be greater than 
for any other point at the same distance from P. 
Draw P P, perpendicular to V p, taking P Pj very 
small; then since '?;=0 in the direction at right 
angles to V p (by the last equation), the velocity of 
P, will equal V, that of P. At P; (since it is nearer 
the central axis of the glacier than P) the rate of increase of V, as depending on y, will 
be less than at P, and therefore yj will also be less (art. 55) ; and for a like reason* (since 
P, is nearer the lower surface than P) yJ will be increased. For both these reasons 
tan"'^^ will be greater at Pj than at P. Draw Pj^, accordingly. Again, draw 
Pi Pa perpendicular to Pi^i, making Pi Pg very small. The velocity of Pg will still = V, 
and the relative velocity oip^ and Pa will be a maximum in the same sense as before. 
We may proceed in the same manner with any number of points. Now, when we pass 
to the limit, by taking P Pi, Pj Pa, &c. indefinitely small, the locus of P Pi Pg, &c. will 
become a continuous curve, possessing the dynamical property, in the case of a glacier, 
that each of its particles moves with the same velocity perpendicular to the plane of the 
paper, and they have consequently no tendency to separate from each other. 
Again, taking § indefinitely small, we shall have another continuous curve ppiP%^ &c. 
contiguous to the former, and so related to it that the relative velocity of two particles 
situated respectively on these curves, and on a common normal to them, vnll be a 
* The increase in the rate at which V increases as we descend from one longitudinal line of motion to 
another, cannot he made a matter of observation, but the analogy with the variation of V in passing from 
the axis of the glacier to its sides (in which case we know that the rate of variation increases) would seem 
fully to justify the assumption. The conclusion of the text, however, is not dependent upon it. 
