76 Mr. Sylvester's Analytical Development 



Also it is clear that rpq=apb=fj^. 

 And to find the inclination of O P to 

 R P Q, we have only to describe a 

 sphere of which P is the centre, and in- 

 ^' tersecting P Q, P R, P O in Q', R', O'. 



Then F O' Q' = /it, and O' Q' = O' R' 



Draw O' N perpendicular to R' Q', then O' N measures the in- 

 clination of the radius vector to the tangent plane *. 



And Q' CK N = 4 



.-.cos 4 =tanO'N.cotO'Q' 



•. cotO'N = 



cot . O Q' 



cos -' 



and .-. cot O' N= i r\ [^ - ^ sin ^ . sin (,^ + ,^,), 



let A O B the angle between the optic axes = 2 ^, then by mere 

 trigonometry 



"° "o" = V/ • : 



i V sm /^ . sin i^^ 



.'. the tangent of the inclination between the radius vector and 

 the normal 



\u c* / V g,n i^ . sin i^, 



Q . E . F. 



In like manner the inclination between the same radius 

 vector *ind the normal at the other point of the wave-surface 

 pierced by it 



= i (r.)'(\ - -t) «i" ('/ - '.) • V — ^^ - ^-^ 



'^ ^ '' \ tt* c* / ^ ' '"^ V sm I, . sini^, 



* O' is the projection of the ray and R' O' of the tangent plane. There- 

 fore O' N being perpendicular to R' Q' represents their inclination. 



