526 Mr. R. J. Sowter on 



So that if 6 is the inclination o£ the radius' to the point :c y 



tan 61= ^- .y =^.tan<6. 



' (-,^) 



One section of the beam being assumed elliptical it follows 

 from the equation (1) of the bounding surface that all sections 

 of the beam are elliptical, the circular sections being particular 

 forms. 



Let the lengths of the primary and secondary focal lines 

 be 2 a and 2 b respectively, and let the focal interval be B. 



The equation to the bounding surface of the beam, the 

 origin being transferred to the middle point of the primary 

 focal line^ can be expressed as 



a^{S- ^)2(S2 f^j;2 .2) +52 g2 ^^2 ^2^0^ 



This equation is the equation of the skew surface generated 

 by the movement of a variable ellipse between the focal lines, 

 the axes of the generating ellipses remaining parallel to the 

 focal lines. 



Through any point [xy z) on this skew surface there is 

 one generator or ray, Adz. : — 



a cos 6.^ - 



h sin (b , . 



This ray intercepts the primary and secondary focal lines 

 at distances from the axis of the beam of a cos (b and b sin (f> 

 respectively. 



If 6 is the inclination of the radius to the point xy z 



tan ^= -T^ — r . tan 6. 



a{h — z) ^ 



Any ray or generator of the astigmatic surface has for its 

 equation 



<h = constant = tan~^ -, — 

 ^ by , 



where 7 and e are the intercepts of the ray or generator on 

 the focal lines, the intercepts being measured from the axis 

 of the beam. 



