MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 543 



three horizontal lines drawn through 0, 1, and 2 respectively, 

 ordinate be supposed to move uniformly 

 along the OT of figure 3, the side of 

 the trigon ABC will decrease in geome- 

 trical progression, while, at the same time, 

 it has a uniform angular motion, making 

 one complete revolution when u moves 

 through the distance GT, that is, six times 

 OT. Also the middle point of the trigon 

 will describe a logarithmic curve, of which 



Then, if the 



Ah, 





A 







f> 



// 







A\ 





/\^\ 



D 



\H 



zt\ 





B~~- ~^^^ 



— A __ 









E 



C 



the ordinate is \e l 



Fig. 4. 



Lemma. 



In order to demonstrate the truth of these assertions, let us draw through A 

 the ordinate ADE, upon AE construct the equilateral trigon AFE, and join 

 FD ; then, since AC and AE are divided similarly at H and D, the trigons ABH, 

 HBC are similar to AFD, DFE. Now the square of FD is obviously AE 2 — 

 ED . DA, that is, DE 2 + ED . DA + DA 2 ; and it can be very easily shown that 

 BC 2 = |FD 2 , wherefore 



BC 2 = a{DE 2 + ED . DA + DA 2 } 



On putting for ED and DA their values A* — At and At 

 formula becomes 



At , the above 



BC : 



= 3 [ A* 2 + A* 2 + A* 2 - At- At - At- At- At- At] 



and this again, according to article 29, passes into 



BC 2 = |e-< 



wherefore the side of the equilateral trigon is given by the formula 



EC = Jl . e-H 



so that, while the ordinate moves over the distance OT, the side of the equilateral 

 trigon is diminished in the ratio of e~ iT to unit ; that is just in the ratio which 

 has already been found for the intervals on the ordinates. 



When the plane of the equilateral trigon passes along the line OA, the base 

 BC is horizontal ; as the plane moves from towards V the end B is raised 

 higher than C, and when that plane passes along the ordinate at V the side AC 

 has become upright ; that is to say, the trigon has made one- twelfth part of a 

 revolution. As the plane is moved beyond V, the turning continues, and for the 



VOL. XXIV. PART III. 7 H 



