28 M. G. Kirchhoff on the Relation of the Lateral Contraction 



A more convenient form may thus be given to the expression. 

 In the first diagram, consider the spiral in which the helix cuts 

 the cylinder whose radius is a. Then ,r = #.sin 6, y=a.cos 0, 



H H 



z= . At the crown. = 0. #=0, z— — . When 6 has a 



n n n 



oX SoC 



small value. Bx=a0. £b= , therefore k-= — an. But -k- for 



n oz os 



one continuous spiral, at the crown of the arch, is evidently in 



the diagram = —tan a. Therefore —an= —tan a. Hence the 



distance, below the centre of the ellipse, of the point at which all 



the face-joints produced will meet, is a . tan a . cotan /3. 



It is to be remarked that this investigation is in no way limited 

 to that extent of arch which is conveniently adopted in practice, 

 never exceeding a semicylinder ; but applies geometrically to the 

 entire cylinder. And if tan a. cotan £ exceeds 1, the joints 

 nowhere tend to become horizontal, but are in certain parts of 

 the circumference parallel to the tangent. 



In practice, the angle a must not differ greatly from the com- 

 plement of (5 ; and, in order to divide the obliquity of bearing of 

 surfaces between the crown and the spring of the arch, a must 

 be a little less than that complement. The distance, therefore, 

 below the centre of the ellipse, of the point at which all the 

 face-joints produced will meet, is a little less than a. (cotan /3) 2 . 

 November 27, 1861. 



V. On the Relation of the Lateral Contraction to the Longitudinal 

 Expansion in Rods of Spying Steel. By G. Kirchhoff*. 



[With a Plate.] 

 To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, Heidelberg, October 17, 1861. 



IN the Number of the Philosophical Magazine for Jane 1861, 

 I find the translation of a paper by M. G. Wertheim, the chief 

 subject matter of which consists in a polemic against a memoir 

 of my own published in vol. cviii. of PoggendorfFs Annalen. 

 Allow me to express the wish that you will publish my original 

 memoir in your Journal, so that your readers may be enabled to 

 decide whether the strictures which M. Wertheim has made are 

 founded on fact or not. 



I am, Gentlemen, 



Your obedient Servant, 

 G. Kirchhoff. 



If a homogeneous cylinder, whose elasticity is the same in 



* Translated from Poggeudorffs Annalen, vol. cviii, p, SCO, by Dr. E, 

 Atkinson. 



