312 Ml" Sang on the Construction vf Oblique Arches 



but it would distribute a uniform downward pressure on each 

 horizontal foot ; and, in this way, the foundation \vould be pres- 

 sed on exactly as if the whole weight of mason-work, from the 

 crown of the one arch to the crown of the other, were piled 

 upon it in squared courses. 



On investigating the forms of the joint on a parabolic skew, 

 1 found its plan to be a line of the third order, the double pa- 

 rabola ; that its end elevation is a semi-cubic parabola ; and 

 that its side elevation is another line of the same order. Stu- 

 dents of the higher mathematics will at once recognise the 

 equations of these curves as the results of other inquiries. 

 For the computations of the parts, on account of the regular 

 progression of the different examples, the method explained in 

 mv treatise On the Solution of Equations of All Orders, will be 

 found to afford peculiar facilities. 



Appendix. 



In the precedino^ part of this paper, I have stated the general princi- 

 ciples which ought to regulate the construction of oblique arches. In 

 this the second part, I propose to enter more into detail, and to give the 

 demonstrations of the theorems above laid down. 



The cencral investigation into the stability of a vault would necessarily 

 be complicated, by the peculiarities of the ultimate abutments, and by 

 the assumed directions of the lines of pressure ; for these directions are, 

 within certain limits, arbitrary. For the present purpose, it is enough to 

 consider the case of a vault resting on parallel abutments, cylindroid, 

 and having the lines of pressure contained in vertical planes parallel to 

 each other. 



Let AB, CD, represent the two abutments, HN the crown line, EF 

 and PN the horizontal projections of two of the lines of pressure. 



Of rectangular co-ordi- 

 , nates, let the x be in the di- 

 rection HG, the y in PM, 

 and the s vertically. 



For convenience, also as- 

 sume oblique co-ordinates 

 V along HN, u along NM, 

 and « as before : put also 

 GHN the angle of the skew, 

 =: s. The formulae of con- 

 version will be 



