410 Mr Larmor, On Critical or [May 28, 



with the simple Tables given by Chambers, {Math. Tables, by Bell) 

 we get a refraction at the visible horizon of about 32'* 0"., by using 

 Ivory's Tables we get about 31'' 30". Bessel's form is not available 

 as he introduces tan. zen. dist. which of course at the horizon is 

 infinite. Allowing 9" for horizontal parallax, and knowing my 

 local time exactly, or within a very few seconds, the first limb 

 agreed with theory to 9", the second was 2'* 43" too early. By 

 Ivory the first limb - 21" too late,- the second 2'* 13" too early. 

 By too early, I me. _l that the Sun's limb had traversed that 

 amount of arc more than was to be expected ; and that it was in 

 advance of its theoretical place. I should mention that the four 

 occasions referred to were the only ones on which I was able to 

 observe an exact Sunset in the course of a voyage of perhaps 12,000 

 miles. 



(6) On Critical or " Apparently Neutral" Equilibrium. By 

 J. Larmoe, M.A. 



1. When a solid body is resting on a fixed surface, its equili- 

 brium is stable when its centre of gravity is vertically below the 

 centre of curvature at the point by which it rests, and unstable 

 when vertically above it : when the two points coincide the equi- 

 librium is often said to be apparently neutral, and its real character 

 is discriminated by an analysis of the differentials of higher orders. 

 It may be worth while to trace the origin of this peculiarity, and 

 its practical effect on the nature of the equilibrium in cases which 

 approximate to this critical condition. 



2. Let us take the case of a heavy body symmetrical about 

 two principal planes through its axis 



A B, (one of them the plane of the 



figure), and resting on a horizontal 



plane at A. The evolute of the 



section has a cusp at 0, the centre 



of curvature corresponding to A. Let 



us suppose it to point downwards, so 



that the radius of curvature is a 



minimum at A, and let us suppose 



the centre of gravity O to be a very 



short distance above O. The position of the body is unstable, 



but a stable position exists in immediate proximity on each side, 



in which the tangents from O to the evolute are vertical. We 



see therefore, that when left free the body will oscillate at first 



round its upright position, and will finally settle down in one of 



these two slightly inclined positions. When G moves down to 



0, these two flanking stable positions come nearer to the upright 



position, and finally come up to it, so that the equilibrium is 



