26 The Astronomer Royal on the Direction of the Joints 



x 

 face transversal to the cylinder, - is constant, or the section of 



the helical surface is a straight line directed to the axis. Hence 

 the equation of the helical surface forming a longitudinal joint 



X 



is - = tau (H — n . z) . 



y 



The equation of the plane of the face of the arch is <r=tan/3. %. 



The combination of these two equations gives the face-joint. 



The second enables us to eliminate z from the first, and we have 



x 



- = tan (H— ?z.cotan 8 ,sc). If X be the horizontal coordinate 



y 



measured in the plane of the face, we can, for joints necessarily 

 in the plane of the face, make #=sin /3 . X j and then the equa- 

 tion of the face-joint, by ordinates upon the plane of the face, is 



?/=X. sin /S.cotan (H— w.cos /3.X). 

 This is not the equation to a straight line ; and therefore the 

 face-joint is curved. 



We may, however, determine the direction of the tangent to 

 the face-joint, either where it meets the intrados, or at the middle 

 of the arch-stones' depth, in the following manner: — Let a be the 

 radius of the cylindrical surface which forms the intrados, or of 

 that which passes through the middle of the stones' depth (as 

 the case may be). And let a -{-8a be the radius of a concentric 

 cylindrical surface, of diameter not differing much from the 

 former. Let x } y, z be the ordinates of the point in the face-joint 

 which corresponds to the former cylindrical surface; x + 8x,y + 8y, 

 z + 8z those for the point in the same face-joint corresponding 

 to the latter surface. Then (as above), x=za . sin 0, y=a . cos 0, 



tt a 



z= , #=tan B . z. To find from these the variations o>, 



n n 



8y, Bz, it must be remarked that 6 is to be varied as well as a } be- 

 cause the face-plane cuts the upper part of the helix at a place 

 where the value of z, and consequently the value of 6, are different 

 from those for the intersection with the lower part of the helix. 

 Thus we find, Sx=Sa. sin + a cos 6 . 80 ; 

 8y = 8a. cos — a sin 0.80 ; 



n 



8x = tan /3. 8z. 

 From these the following values are easily found (putting 8a ! 



n 8a 



tor 



,)= 



w#.cos # + tan/3, 



S#=sin #.tan /3.8a' ; 



8y = (na + tan ft . cos 0) . ca! ; 



8&=z sin 0.8a f . . 



