Put (aox 3 -}-a l x 2 +a 2 x +a 3) (box' +b1x'+b2) - (aox'3+aix'2+a2x'+a3) (box' + bix + b2) = 0; after division by x-x the three equations are formed aobcx 2 = aobix+aob2 =0, aobix 2 + (aob2+a1b1-a2bo) x +alb2 -a3bo = 0, aob2x 2 +(a02-a3bo)x+a2b2-a3b1 =0 and thence the resultant aobo ao aob2 aob 1 aob2+a1b1-a2bo alb2-a3b0 aob 2 a1b2 - a 3 bo a2b2 - a3b1 which is a symmetrical determinant.
Let AoBo be a plane wave-surface of the light before it falls upon the prisms, AB the corresponding wave-surface for a particular part of the spectrum after the light has passed the prisms, or after it has passed the eye-piece of the observing telescope.
The path of a ray from the wave-surface AoBo to A or B is determined by the con dition that the optical distance, µ ds, is a minimum; and, as AB is by supposition a wave-surface, this optical distance is the same for both points.
Accordingly, the optical distance from AoBo to A is represented by f (A +S/c)ds, the integration being along the original path Ao.
A; and similarly the optical distance between AoBo and B is represented by f (,t+So.)ds, the integration being along Bo.