Lord Rayleigh's Investigations in Optics. 483 



of the prisms are surfaces of revolution, so that it is possible 

 by proper adjustments to render every thing symmetrical with 

 respect to a plane bisecting at right angles the refracting 

 edges. If p be the primary focal length, this plane is that 

 represented by y = 0, and the equation of the wave-surface 

 reduces to 



*=£ + $+«* z +r^> • • • • ( 2 ) 



terms of higher order being omitted. 



The constants « and 7 in (2) may be interpreted in terms 

 of the differential coefficients of the principal radii of curva- 

 ture. By the usual formula, the radius of curvature at the 

 point x of the intersection of (2) with the plane y = is ap- 

 proximately p(l — 6otpx). Since y = is a principal plane 

 throughout, this radius of curvature is a principal radius of 

 the surface ; so that, denoting it by p, we have 



a= ~W' C dx • • '• • • ■ • (3) 

 In the neighbourhood of the origin the approximate value of 

 the product of the principal curvatures is 



Thus 



1 6ux 2yx 



/ T* T~ \ * 



pp p p 



f 1 \ _ Bp Bp f _ 6xx 2yx 



whence by (3), 



1 dp' 



^~w^ (4) 



The equation of the normal at the point x, y, z is 



b p- l x + 2>oLX 2 + yy 2 p'- 1 y + 2yxy' ' ' W 



and its intersection with the plane ^-=p occurs at the point 

 determined approximately by 



f=-p(3«^ 2 + 7y 2 ),l 



P'-P 9 I (6) 



V =£— r ^-2pyxy,J 



r 

 terms of the third order being omitted. 



According to geometrical optics, the thickness of the image 

 of a luminous line at the primary focus is determined by the 

 extreme value off ; and for good definition in the spectroscope 



