MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 533 



the length of the arc AB must be cat t. Now the tangent of the inclination at 

 B is also cat t, and it is the property of all arches that the tangent of the incli- 

 nation is proportional to the weight reckoned from the horizontal part of the 

 curve, and consequently the curve of which the ordinate is sus t is that which a 

 flexible chain assumes. 



Since the ordinate TB is the derivative of the area OABT regarded as a 

 function of OT, that area must also be proportional to the function cat t ; that is 

 to say, the surface AOTB is proportional to the length of the curve AB. 



21. If along the line TB we make TC proportional to the conjugate function 

 cat t, the point C is in a curve C'OC, which we may call the companion to the 

 catenary ; this curve crosses the axis at the origin O, so that at the distance — t 

 the ordinate TC appears on the opposite side ; and the area OTC is proportional 

 to HB, the excess of TB above the linear unit A. The positive branches of these 

 two curves approach more and more closely as t is taken of greater value. 



22. By making TD a third proportional to TB and TH, we obtain the repre- 

 sentative of the function which, for want of a better notation, we have indicated 

 by the symbol sec t. The curve traced by the point D rises to touch the catenary 

 at the point A, and approaches on the positive and negative sides to the line of 

 abscissae. When the arc AB is unfolded, the extremity of the tangent describes, 

 as is well known, the line called the Tractory ; now if, on the surface of an 

 oblique circular arch, a line be drawn crossing all the lines of pressure at right 

 angles, this line indicates the proper course for the joints of the voussoirs; it is a 

 line of double curvature, and I have shown, in a paper on the construction of 

 oblique arches, read before the Society of Arts for Scotland in 1835 {Edin. New 

 Phil. Jour, for April 1840), that the projection of this line upon the plane of the 

 parapet is the tractory, while the projection of the same line upon a plane crossing 

 the roadway at right angles is a modification of the curve D'AD ; on that account 

 I have called it the companion to the tractory. 



23. If we now make TE a third proportional to TC and TH, TE becomes the 

 representative of the function cse t. The curve traced by E has the continuation 

 of OA for one asymptote, and the line of abscissas for another. For negative 

 values of t it appears on the opposite sides of those lines, the two branches of this 

 curve being disconnected. It crosses the companion to the catenary at t where 

 that line crosses the parallel to OT through A. 



24. Making TB : TC : : TH : TF, and TC : TB : : TH : TG we obtain TF the 

 representative of tan t, and TG that of cot t. The line of tangents osculates and 

 crosses the companion to the catenary at 0, and has the lines AH and A'H for 

 asymptotes ; and it is remarkable that this line, to which in the abovementioned 

 paper 1 have given the name double-logarithmic, is the projection of the same line 

 of double curvature upon the horizontal plane. The curve of cotangents, traced 

 by the points G and G', consists of two detached branches, the positive branch 



