292 



Mr. E. J. Reed on 'the Distribution 



[Feb. 9, 



III. "On the Problem of the In- and Circumscribed Triangle." By 



A. Cayley/E.R.S. Received December 30, 1870, 



(Abstract.) ' 



The problem of the in and circumscribed triangle is a particular case of 

 that of the in- and circumscribed polygon : the last-mentioned problem may 

 be thus stated — to find a polygon such that the angles are situate in and 

 the sides touch a given curve or curves. And we may in the first instance 

 inquire as to the number of such polygons. In the case where the curves 

 containing the angles and touched by the sides respectively are all of them 

 distinct curves, the number of polygons is obtained very easily and has a 

 simple expression : it is equal to twice the product of the orders of the curves 

 containing the several angles respectively into the product of the classes of 

 the curves touched by the several sides respectively ; or, say, it is equal to 

 twice the product of the orders of the angle-curves into the product of the 

 classes of the side-curves. But when several of the curves become one and 

 the same curve, and in particular when the angles are all of them situate 

 in and the sides all touch one and the same curve, it is a much more diffi- 

 cult problem to find the number of polygons. The solution of this problem 

 when the polygon is a triangle, and for all the different relations of identity 

 between the different curves, is the object of the present memoir, which is 

 accordingly entitled " On the Problem of the In- and Circumscribed Tri- 

 angle ; " the methods and principles, however, are applicable to the case 

 of a polygon of any number of sides, the method chiefly made use of being 

 that furnished by the theory of correspondence, 



IV. "On the Unequal Distribution of Weight and , Support in 

 Ships, audits Effects in Still Wafer, in Waves, and in Exceptional 

 Positions on Shore." By E. J. Reed, C ? B., Vice-President of 

 the Institution of Naval Architects. Communicated by Prof. 

 G. G. Stokes, Sec. R.S. Received December 31, 1870. 



(Abstract.) 



The object of this paper is to bring within the grasp of calculation what 

 the author considers a much neglected division of shipbuilding science and 

 art, by investigating the actual longitudinal bending- and shearing-strains to 

 which the structure is exposed in ships of various forms under the varying 

 conditions to which all ships are more or less liable. The weakness exhi- 

 bited by many ships has long pointed to the necessity of further investiga- 

 tion in this direction ; and two modern events (the use of iron and steel in 

 shipbuilding, and the introduction of armoured ships) have added much to 

 the urgency of the inquiry. 



After glancing briefly at the state of the question as presented in the 

 writings of Bouguer, Bernoulli, Euler, Don Juan D'Ulloa, Romme, Dupin, 



