56 Mr. J. M. Heppel on the Theory of Continuous Beams, 



lowing manner : — If the curve determined by the equation V=0 be traced 

 in the first quadrant, every straight line parallel to the axis of p meets the 

 curve once, and only once, and every straight line parallel to the axis of -, 

 and at a greater distance from it than _p , meets the curve once, and only 



dY 



once. As -j- vanishes with r, the curve meets the axis of p at right angles ; 



and from Art. 15 it follows that when r and _p are indefinitely great, 

 the angle which the tangent to the curve makes with the axis of p is very 

 nearly a right angle. Thus, for small values of r the curve is concave to 

 the axis of and for very large values of r the curve is convex to the axis 

 of^> ; so that the curve must have a point or points of inflexion. 



24. In the very careful account of Ivory's mathematical researches, 

 which is given in the fourth volume of the 'Abstracts of the Papers ... of 

 the Royal Society,' it is said, with respect to Jacobi's theorem, "In a 

 paper in the Transactions for 1838, Mr. Ivory has with great elegance 

 demonstrated this theorem, and has given, with greater detail than its 

 authors had entered on, several statements regarding the limitations of the 

 proportions of the axes." The language is cautious, but seems to imply 

 some suspicion with regard to the accuracy of the statements. As we 

 have now seen, many of Ivory's statements are inaccurate, and others, 

 though accurate, are based on unsound reasoning. 



"On the Theory of Continuous Beams." By John Mortimer 

 Heppel, M. Inst. C.E. Communicated by Prof. W. J. Mac- 

 qtjorn Rankine. Received December 9, 1869*. 



In venturing to present to the Royal Society a paper on a subject which 

 has engaged the attention, more especially in France, of some of the most 

 eminent engineers and writers on Mechanical Philosophy, the author feels 

 it to be incumbent on him to state the nature of the claim to their attention 

 which he hopes it may be found to possess in point of originality or im- 

 provement on the method of treatment. 



To do this clearly, however, it will be necessary to advert to the 

 principal steps by which progress in the knowledge of this subject has been 

 made, both in France and in this country. 



The theory of continuous beams appears to have first attracted attention 

 in France about 1825, when a method of determining all the conditions 

 of equilibrium of a straight beam of uniform section throughout, resting 

 on any number of level supports at any distances apart, each span being 

 loaded uniformly, but the uniform loads varying in any manner from one 

 span to another, was investigated and published by M. Navier. This 

 method, although perfectly exact for the assumed conditions, was objection- 

 able from the great labour and intricacy of the calculations it entailed. 



* Bead January 27, 1870. See vol. xviii. p. 176. 



