On Non-homocentric Pencils. 711 
From the second of these equations values of + were com- 
puted for different values of 0, in the planes z==0 and z=c. 
Sections of the pencil by these two planes were obtained 
by drawing a number of concentric circles with the values 
of r for radii, crossed by radial lines at intervals of 5°. It 
will be obvious that corresponding points on the two sections 
lie on the same ray. The numerical values taken in the 
preparation of fig. 5 are: a=20, b=—10°4, c= —5, while 
d has values rising from 0 to ‘1 by ‘Ol ata time. Parts of 
the two sections obtained are reproduced in fig. 5. The 
section at z= —10'4 was drawn to 20 times the scale of the 
other. 



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(2= 20 
Sadion Ol X=-10-4 



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A thread was stretched over the first section to represent 
the position of the wire, the angular positions of the inter- 
sections with the wire were noted, and the corresponding 
points on the second section were marked on a piece of 
tracing-paper placed upon it. The form of the shadow is 
obtained by drawing through these points. The results are 
