1862.] 305 



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 ; arid 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 

 actual applications it is necessary to determine some remaining 

 constants by considerations peculiar to each case. Now there is one 

 modification of the strains whose value can be ascertained by ordinary 

 mechanics, namely, the horizontal part of compressive force in the 

 part of the beam above the neutral line, and the horizontal part of 

 tension force in the part of the beam below the neutral line. (These 

 words apply to a beam supported at both ends ; in the case of a beam 

 projecting from a wall, the words " compression " and "tension" 

 must be reversed.) By determining the corresponding expression 

 on the theory of this memoir, and comparing the two, the remaining 

 constants and the form of the function are completely determined. 

 From its partial differential coefficients are found the three functions 

 depending on the forces acting at any one point (as already men- 



z 2 



