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A “ smith’s PRIZE ” PAPER ; SOLUTIONS. 
61—2 
But if u be the eccentric anomaly, then 
x = a cos u, y - b sin u, = a f(l — e 2 ) sin u, 
and the equations become 
- — Sill u 
a? (1 — e cos u) 3 
H sm u 
and multiplying by — cos u, — sin u respectively, and adding, we have 
which is the required differential relation. 
9. Show that the attraction of an indefinitely thin double-convex lens on a 'point at 
the centre of one of its faces is equal to that of the infinite plate included between the 
tangent plane at the point and the parallel tangent plane of the other face of the lens. 
The figure represents the upper half only of the lens, but in speaking of any 
portion thereof, such as PRQ, we include the symmetrically situate portion of the 
under-half of the lens. 
Let a, = PQ, be the thickness of the lens, Z NPQ = A, which angle is ultimately 
= 2' r ^'^ ien it i s a t once seen that the attraction of the cone NPQ is = 27ra(l — cos X): 
and from this it follows that the attraction of the infinite plate is = 27ra. The 
attraction of the whole infinite plate except the cone NPQ is = 27ra cos X, which is 
indefinitely small in regard to 27ra; and, a fortiori, the attraction of the portion MPR 
of the lens is indefinitely small in regard to 27ra. We have then only to show that 
the attraction of the solid NRQ is indefinitely small in regard to 27ra; for, this being 
so, the attraction of the lens may be taken to be equal to that of the cone NPQ, 
and will therefore ultimately be = 27ra, the attraction of the infinite plate.