COLOURS OF THICK PLATES. [151] 



the direction of as positive. Let a, b, c be the co-ordinates of L ; a, b', c those of M; and 

 suppose a, b, a, and b' small compared with c, c, r, and s. Let x, y, z be the co-ordinates 

 of any point P on the dimmed surface, R the retardation of the stream which was scattered 

 at P on emergence, relatively to that which was scattered at the same point on entrance. Let 

 L t , L«, L 3 be the images of L after refraction, reflexion, and second refraction, respectively ; 

 M it M ti M 3 , the images of M; and let a, b, c, or a , b', c, with the suffixes 1, 2, 3, denote the 

 co-ordinates of L u L s , L 3 , or M u M 2 , M 3 . In approximating to the value of R, let the squares 

 of the small quantities a, b, x, y, &c. be retained, so that the terms neglected are of the fourth 

 order, since all the terms are of even orders, as will be immediately seen when the approximation 

 is commenced. 



2. The rays diverging from L may after refraction be supposed to diverge from L u not- 

 withstanding the spherical aberration of direct pencils, and the astigmatism of oblique pencils. 

 For, first, let L be in the axis. The supposition that the rays diverge from L x is equivalent to 

 supposing that the front of a wave is a sphere having Z, for centre, whereas it is really a 

 surface of revolution such that L x is the centre of curvature of a section made by a plane 

 through the axis. This plane cuts the sphere above mentioned in a circle, which, being a circle 

 of curvature, cannot have with the curve a contact lower than one of the second order. But 

 the contact is actually of the third order, since the curve and circle touch without cutting. 

 Hence the error produced in the calculation of R by supposing the front of a wave to be a 

 sphere, instead of that surface which it actually is, is only a small quantity of the fourth order, 

 and quantities of this order are supposed to be neglected. 



Next, consider an oblique pencil. Let L' and L" be two points in the axis of the pencil 

 which are the centres of curvature of its principal sections. If the distance of L' and L" from 

 each other, and from Z,„ were not small, the front of the wave would have a contact of the 

 first order with a sphere described round L x , with such a radius as to pass through the point 

 where the front is cut by the axis of the pencil ; and in that case the error committed by 

 taking the sphere for the actual front would be of the second order. But L' and L" are 

 situated at distances from L y which are small quantities of the second order, whence it will 

 readily be seen that the actual error is only of the fourth order. 



3. Let the expression (Z, to L 3 ) denote the retardation of a wave proceeding from L to 

 L 3 , or rather, in case L 3 be a virtual focus, the difference of retardations of two waves starting 

 from L and L 3 and reaching the same given point. Then 



R = (L to L 3 ) + PM- PL 3 - {{M to M 3 ) + PL - PM 3 \ = K + V, 

 where 



K = (L to L 3 ) - (M to M 3 ), 



V = PM- PL 3 - {PL - PM 3 ) = r - V", suppose. 



According to the explanation given in the preceding article, when the position of P changes 

 K remains constant, to the degree of approximation which it is proposed to employ, but the 

 value of V depends upon the position of P. We have 



