150 Royal Society : — 



The theory contemplates forces acting in one plane. A beam, 

 therefore, is considered as a lamina in a vertical plane, — the same 

 considerations applying to every vertical lamina of which a beam 

 may be conceived to be composed. 



The author remarks that it is unnecessary to recognize every 

 possible strain in a beam. Metallic masses are usually in a state of 

 strain from circumstances occurring in their formation ; but such 

 strains are not the subject of the present investigation, which is 

 intended to ascertain only those strains which are created by the 

 weight of the beam and its loads. The algebraical interpretation of 

 this remark is, that it is not necessary to retain general solutions of 

 the equations which will result from the investigation, but only such 

 solutions as will satisfy the equations. 



After defining the unit of force as the weight of a square unit of 

 the lamina, and the measure of compression-thrust or extension-pull 

 as the length of the ribbon of lamina whose breadth is the length of 

 the line which is subject to the transverse action of the compression 

 or tension, and whose weight is equal to that compression or tension, 

 the author considers the effect of tension, &c, estimated in a direction 

 inclined to the real direction of the tension, and shows that it is 

 proportional to the square of the cosine of inclination. He then 

 considers the effect of compounding any number of strains of 

 compression or tension which may act simultaneously on the same 

 part of a lamina, and shows that their compound effect may, in every 

 case, be replaced by the compound effect of two forces at right 

 angles to each other, the two forces being both compressions, or 

 both tensions, or one compression and one tension. Succeeding 

 investigations are therefore limited to two such forces. 



Proceeding then to the general theory of beams, it is remarked 

 that if a curve be imagined, dividing a beam into any two parts, the 

 further part of the beam (as estimated from the origin of coordinates) 

 may be considered to be sustained by the forces which act in various 

 directions across that curve, taken in combination with the weight of 

 the further part of the beam, the load upon that part, the reaction 

 of supports, &c. Expressing the forces in conformity with the 

 principles already explained, the three equations of equilibrium are 

 formed, in which are involved several integrals depending on the 

 form of the curve and on the forces. As the same equations must 

 apply to any adjacent curve, the author remarks that this is a proper 

 case for application of the Calculus of Variations ; and on making that 

 application, a remarkable relation is found to exist among the three 

 functions depending on the forces acting at one point, from which it 

 is immediately inferred that their algebraical expressions are the 

 partial differential coefficients (of the second order) of a single 

 function of the coordinates of the point of action. On substituting 

 the partial differential coefficients, the integrations can be immediately 

 performed ; and the three equations assume a form of great simplicity, 

 from which the sign of integration has entirely disappeared. 



A form is then assumed for the principal function, with inde- 

 terminate coefficients, and it is shown that some of the constants 

 may be eliminated by means of the three equations. But in the 



